67 results

1, 9, 25, 49, .......... What is next

1, 9, 25, 49, .......... What is next
1^2 = 1
3^2 = 9
5^2 = 25
7^2 = 49
So this pattern takes odd numbers and squares them. Our next odd number is 9:
9^2 = [B]81[/B]

10 times the square of a number w divided by 12

10 times the square of a number w divided by 12
The square of a number w
w^2
10 times this
10w^2
Divided by 12
[B]10w^2/12[/B]

15 less than a number squared

15 less than a number squared
A number is denoted by an arbitrary variable, let's call it x.
x
Squared means we raise that number to a power of 2
x^2
15 less means we subtract
[B]x^2 -15[/B]

3 is subtracted from square of a number

3 is subtracted from square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Square of a number means we raise x to the 2nd power:
x^2
3 is subtracted from square of a number
[B]x^2 - 3[/B]

3 times the square of a number x minus 12

3 times the square of a number x minus 12.
Build the algebraic expression piece by piece:
[LIST]
[*]Square of a number x: x^2
[*]3 times this: 3x^2
[*]Minus 12: [B]3x^2 - 12[/B]
[/LIST]

30 increased by 3 times the square of a number

30 increased by 3 times the square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x
x
The square of a number means we raise x to the power of 2:
x^2
3 times the square:
3x^2
The phrase [I]increased by[/I] means we add 3x^2 to 30:
[B]30 + 3x^2[/B]

30 increased by 3 times the square of a number

Let "a number" equal the arbitrary variable x.
The square of that is x^2.
3 times the square of that is 3x^2.
Now, 30 increased by means we add 3x^2 to 30
30 + 3x^2

5 squared minus a number x

5 squared minus a number x
5 squared is written as 5^2
Minus a number x means we subtract the variable x
[B]5^2 - x[/B]

6 is divided by square of a number

6 is divided by square of a number
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
x
the square of this means we raise x to the power of 2:
x^2
Next, we divide 6 by x^2:
[B]6/x^2[/B]

A bag of fertilizer covers 300 square feet of lawn. Find how many bags of fertilizer should be purch

A bag of fertilizer covers 300 square feet of lawn. Find how many bags of fertilizer should be purchased to cover a rectangular lawn 290 feet by 150 feet.
The area of a rectangle is length * width, so we have:
A = 290 * 150
A = 43,500 sq ft.
Now, to find the number of bags needed for a 300 square feet per bag of fertilizer, we have:
Bags Needed = Total Square Feet of Lawn / Square Feet covered per bag
Bags Needed = 43,500 / 300
Bags Needed = [B]145[/B]

A triangular garden has base of 6 meters amd height of 8 meters. Find its area

A triangular garden has base of 6 meters amd height of 8 meters. Find its area
Area (A) of a triangle is:
A = bh/2
Plugging in our numbers, we get:
A = 6*8/2
A = [B]24 square meters[/B]

Babylonian Method

Free Babylonian Method Calculator - Determines the square root of a number using the Babylonian Method.

Basic Statistics

Free Basic Statistics Calculator - Given a number set, and an optional probability set, this calculates the following statistical items:

Expected Value

Mean = μ

Variance = σ^{2}

Standard Deviation = σ

Standard Error of the Mean

Skewness

Mid-Range

Average Deviation (Mean Absolute Deviation)

Median

Mode

Range

Pearsons Skewness Coefficients

Entropy

Upper Quartile (hinge) (75th Percentile)

Lower Quartile (hinge) (25th Percentile)

InnerQuartile Range

Inner Fences (Lower Inner Fence and Upper Inner Fence)

Outer Fences (Lower Outer Fence and Upper Outer Fence)

Suspect Outliers

Highly Suspect Outliers

Stem and Leaf Plot

Ranked Data Set

Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range

Root Mean Square

Weighted Average (Weighted Mean)

Frequency Distribution

Successive Ratio

Expected Value

Mean = μ

Variance = σ

Standard Deviation = σ

Standard Error of the Mean

Skewness

Mid-Range

Average Deviation (Mean Absolute Deviation)

Median

Mode

Range

Pearsons Skewness Coefficients

Entropy

Upper Quartile (hinge) (75th Percentile)

Lower Quartile (hinge) (25th Percentile)

InnerQuartile Range

Inner Fences (Lower Inner Fence and Upper Inner Fence)

Outer Fences (Lower Outer Fence and Upper Outer Fence)

Suspect Outliers

Highly Suspect Outliers

Stem and Leaf Plot

Ranked Data Set

Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range

Root Mean Square

Weighted Average (Weighted Mean)

Frequency Distribution

Successive Ratio

Complex Number Operations

Free Complex Number Operations Calculator - Given two numbers in complex number notation, this calculator:

1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.

2) Determines the Square Root of a complex number denoted as √a + bi

3) Absolute Value of a Complex Number |a + bi|

4) Conjugate of a complex number a + bi

1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.

2) Determines the Square Root of a complex number denoted as √a + bi

3) Absolute Value of a Complex Number |a + bi|

4) Conjugate of a complex number a + bi

difference between 2 positive numbers is 3 and the sum of their squares is 117

difference between 2 positive numbers is 3 and the sum of their squares is 117
Declare variables for each of the two numbers:
[LIST]
[*]Let the first variable be x
[*]Let the second variable be y
[/LIST]
We're given 2 equations:
[LIST=1]
[*]x - y = 3
[*]x^2 + y^2 = 117
[/LIST]
Rewrite equation (1) in terms of x by adding y to each side:
[LIST=1]
[*]x = y + 3
[*]x^2 + y^2 = 117
[/LIST]
Substitute equation (1) into equation (2) for x:
(y + 3)^2 + y^2 = 117
Evaluate and simplify:
y^2 + 3y + 3y + 9 + y^2 = 117
Combine like terms:
2y^2 + 6y + 9 = 117
Subtract 117 from each side:
2y^2 + 6y + 9 - 117 = 117 - 117
2y^2 + 6y - 108 = 0
This is a quadratic equation:
Solve the quadratic equation 2y2+6y-108 = 0
With the standard form of ax2 + bx + c, we have our a, b, and c values:
a = 2, b = 6, c = -108
Solve the quadratic equation 2y^2 + 6y - 108 = 0
The quadratic formula is denoted below:
y = -b ± sqrt(b^2 - 4ac)/2a
[U]Step 1 - calculate negative b:[/U]
-b = -(6)
-b = -6
[U]Step 2 - calculate the discriminant Δ:[/U]
Δ = b2 - 4ac:
Δ = 62 - 4 x 2 x -108
Δ = 36 - -864
Δ = 900 <--- Discriminant
Since Δ is greater than zero, we can expect two real and unequal roots.
[U]Step 3 - take the square root of the discriminant Δ:[/U]
√Δ = √(900)
√Δ = 30
[U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U]
Numerator 1 = -b + √Δ
Numerator 1 = -6 + 30
Numerator 1 = 24
[U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U]
Numerator 2 = -b - √Δ
Numerator 2 = -6 - 30
Numerator 2 = -36
[U]Step 6 - calculate your denominator which is 2a:[/U]
Denominator = 2 * a
Denominator = 2 * 2
Denominator = 4
[U]Step 7 - you have everything you need to solve. Find solutions:[/U]
Solution 1 = Numerator 1/Denominator
Solution 1 = 24/4
Solution 1 = 6
Solution 2 = Numerator 2/Denominator
Solution 2 = -36/4
Solution 2 = -9
[U]As a solution set, our answers would be:[/U]
(Solution 1, Solution 2) = (6, -9)
Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution

Estimate Square Roots

Free Estimate Square Roots Calculator - Estimates the square root of a number

Explain the relationship between "squaring" a number and finding the "square root" of a number. Use

Explain the relationship between "squaring" a number and finding the "square root" of a number. Use an example to further explain your answer.
Squaring a number means raising it to the power of 2
The square root of a number [I]undoes[/I] a square of a number.
So square root of x^2 is x
x squared is x^2
Let x = 5.
x squared = 5^2 = 25
Square root of 25 = square root of 5^2 = 5

Factoring and Root Finding

Free Factoring and Root Finding Calculator - This calculator factors a binomial including all 26 variables (a-z) using the following factoring principles:

* Difference of Squares

* Sum of Cubes

* Difference of Cubes

* Binomial Expansions

* Quadratics

* Factor by Grouping

* Common Term

This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots

* Factors and simplifies Rational Expressions of one fraction

* Determines the number of potential*positive* and *negative* roots using Descarte’s Rule of Signs

* Difference of Squares

* Sum of Cubes

* Difference of Cubes

* Binomial Expansions

* Quadratics

* Factor by Grouping

* Common Term

This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots

* Factors and simplifies Rational Expressions of one fraction

* Determines the number of potential

Hari planted 324 plants in such a way that there were as many rows of plants as there were number of

Hari planted 324 plants in such a way that there were as many rows of plants as there were number of columns. Find the number of rows and columns.
Let r be the number of rows and c be the number of columns. We have the area:
rc = 324
Since rows equal columns, we have a square, and we can set r = c.
c^2 = 324
Take the square root of each side:
[B]c = 18[/B]
Which means [B]r = 18[/B] as well.
What we have is a garden of 18 x 18.

if a number is added to its square, it equals 20

if a number is added to its square, it equals 20.
Let the number be an arbitrary variable, let's call it n.
The square of the number means we raise n to the power of 2:
n^2
We add n^2 to n:
n^2 + n
It equals 20 so we set n^2 + n equal to 20
n^2 + n = 20
This is a quadratic equation. So [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn%3D20&pl=Solve+Quadratic+Equation&hintnum=+0']we type this equation into our search engine[/URL] to solve for n and we get two solutions:
[B]n = (-5, 4)[/B]

if a number is added to its square, the result is 72. find the number

if a number is added to its square, the result is 72. find the number.
Let the number be n. We're given:
n + n^2 = 72
Subtract 72 from each side, we get:
n^2 + n - 72 = 0
This is a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-72%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this equation into our search engine[/URL], and we get:
[B]n = 8 and n = -9[/B]

It took 3.5 gallons of paint to cover a wall that is 985 square feet. How many gallons will it take

It took 3.5 gallons of paint to cover a wall that is 985 square feet. How many gallons will it take to cover a wall that is 6501 square feet?
Set up a proportion of gallons of paint to square feet where n is the number of gallons of paint to cover 6501 square feet
3.5/985 = n/6501
Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=3.5&num2=n&den1=985&den2=6501&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL], we get:
n = [B]23.1[/B]

Jennifer is playing cards with her bestie when she draws a card from a pack of 25 cards numbered fro

Jennifer is playing cards with her bestie when she draws a card from a pack of 25 cards numbered from 1 to 25. What is the probability of drawing a number that is square?
The squares from 1 - 25 less than or equal to 25 are as follows:
[LIST=1]
[*]1^2 = 1
[*]2^2 = 4
[*]3^2 = 9
[*]4^2 = 16
[*]5^2 = 25
[/LIST]
So the following 5 cards are squares:
{1, 4, 9, 16, 25}
Therefore, our probability of drawing a square is:
P(square) = Number of Squares / Number of Cards
P(square) = 5/25
This fraction can be simplified. So [URL='https://www.mathcelebrity.com/fraction.php?frac1=5%2F25&frac2=3%2F8&pl=Simplify']we type in 5/25 into our search engine, choose simplify[/URL], and we get:
P(square) = [B]1/5[/B]

Lagrange Four Square Theorem (Bachet Conjecture)

Free Lagrange Four Square Theorem (Bachet Conjecture) Calculator - Builds the Lagrange Theorem Notation (Bachet Conjecture) for any natural number using the Sum of four squares.

Let n be an integer. If n^2 is odd, then n is odd

Let n be an integer. If n^2 is odd, then n is odd
Proof by contraposition:
Suppose that n is even. Then we can write n = 2k
n^2 = (2k)^2 = 4k^2 = 2(2k) so it is even
[I]So an odd number can't be the square of an even number. So if an odd number is a square it must be the square of an odd number.[/I]

m is inversely proportional to the square of p-1 when p=4 m=5 find m when p=6

m is inversely proportional to the square of p-1 when p=4 and m=5. find m when p=6
Inversely proportional means there is a constant k such that:
m = k/(p - 1)^2
When p = 4 and m = 5, we have:
5 = k/(4 - 1)^2
5 = k/3^2
5 = k/9
[U]Cross multiply:[/U]
k = 45
[U]The problems asks for m when p = 6. And we also now know that k = 45. So plug in the numbers:[/U]
m = k/(p - 1)^2
m = 45/(6 - 1)^2
m = 45/5^2
m = 45/25
m = [B]1.8[/B]

Nine workers are hired to harvest potatoes from a field. Each is given a plot which is 5x5 feet in s

Nine workers are hired to harvest potatoes from a field. Each is given a plot which is 5x5 feet in size. What is the total area of the field?
Area of each plot is 5x5 = 25 square feet.
Total area = Area per plot * number of plots
Total area = 25 sq ft * 9
Total area = [B]225 sq ft[/B]

Number Property

Free Number Property Calculator - This calculator determines if an integer you entered has any of the following properties:

* Even Numbers or Odd Numbers (Parity Function or even-odd numbers)

* Evil Numbers or Odious Numbers

* Perfect Numbers, Abundant Numbers, or Deficient Numbers

* Triangular Numbers

* Prime Numbers or Composite Numbers

* Automorphic (Curious)

* Undulating Numbers

* Square Numbers

* Cube Numbers

* Palindrome Numbers

* Repunit Numbers

* Apocalyptic Power

* Pentagonal

* Tetrahedral (Pyramidal)

* Narcissistic (Plus Perfect)

* Catalan

* Repunit

* Even Numbers or Odd Numbers (Parity Function or even-odd numbers)

* Evil Numbers or Odious Numbers

* Perfect Numbers, Abundant Numbers, or Deficient Numbers

* Triangular Numbers

* Prime Numbers or Composite Numbers

* Automorphic (Curious)

* Undulating Numbers

* Square Numbers

* Cube Numbers

* Palindrome Numbers

* Repunit Numbers

* Apocalyptic Power

* Pentagonal

* Tetrahedral (Pyramidal)

* Narcissistic (Plus Perfect)

* Catalan

* Repunit

One fifth of the square of a number

One fifth of the square of a number
We have an algebraic expression. Let's break this into parts.
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]The square of a number means we raise it to the power of 2. So we have x^2
[*]One-fifth means we have a fraction, where we divide our x^2 in Step 2 by 5. So we get our final answer below:
[/LIST]
[B]x^2/5[/B]

One number is equal to the square of another. Find the numbers if both are positive and their sum is

One number is equal to the square of another. Find the numbers if both are positive and their sum is 650
Let the number be n. Then the square is n^2. We're given:
n^2 + n = 650
Subtract 650 from each side:
n^2 + n - 650 = 0
We have a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-650%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this into our search engine[/URL] and we get:
n = 25 and n = -26
Since the equation asks for a positive solution, we use [B]n = 25[/B] as our first solution.
the second solution is 25^2 = [B]625[/B]

Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job

Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job.
Let y be the number of square yards. We have the following equation:
0.2y + 10 = 25
To solve for y, we[URL='https://www.mathcelebrity.com/1unk.php?num=0.2y%2B10%3D25&pl=Solve'] type this equation into our search engine [/URL]and we get:
y = [B]75[/B]

Prove sqrt(2) is irrational

Use proof by contradiction. Assume sqrt(2) is rational.
This means that sqrt(2) = p/q for some integers p and q, with q <>0.
We assume p and q are in lowest terms.
Square both side and we get:
2 = p^2/q^2
p^2 = 2q^2
This means p^2 must be an even number which means p is also even since the square of an odd number is odd.
So we have p = 2k for some integer k. From this, it follows that:
2q^2 = p^2 = (2k)^2 = 4k^2
2q^2 = 4k^2
q^2 = 2k^2
q^2 is also even, therefore q must be even.
So both p and q are even.
This contradicts are assumption that p and q were in lowest terms.
So sqrt(2) [B]cannot be rational.
[MEDIA=youtube]tXoo9-8Ewq8[/MEDIA][/B]

Rational,Irrational,Natural,Integer Property

Free Rational,Irrational,Natural,Integer Property Calculator - This calculator takes a number, decimal, or square root, and checks to see if it has any of the following properties:

* Integer Numbers

* Natural Numbers

* Rational Numbers

* Irrational Numbers Handles questions like: Irrational or rational numbers Rational or irrational numbers rational and irrational numbers Rational number test Irrational number test Integer Test Natural Number Test

* Integer Numbers

* Natural Numbers

* Rational Numbers

* Irrational Numbers Handles questions like: Irrational or rational numbers Rational or irrational numbers rational and irrational numbers Rational number test Irrational number test Integer Test Natural Number Test

s = tu^2 for u

s = tu^2 for u
Divide each side by t
u^2 = s/t
Take the square root of each side
[LIST]
[*]u = sqrt(s/t)
[*]u = -sqrt(s/t)
[/LIST]
We have two answers due to negative number squared is positive

Square Number

Free Square Number Calculator - This calculator determines the nth square number

Square Root Table

Free Square Root Table Calculator - Generates a square root table for the first (n) numbers rounded to (r) digits

Square Roots and Exponents

Free Square Roots and Exponents Calculator - Given a number (n), or a fraction (n/m), and/or an exponent (x), or product of up to 5 radicals, this determines the following:

* The square root of n denoted as √n

* The square root of the fraction n/m denoted as √n/m

* n raised to the x^{th} power denoted as n^{x} (Write without exponents)

* n raised to the x^{th} power raised to the yth power denoted as (n^{x})^{y} (Write without exponents)

* Product of up to 5 square roots: √a√b√c√d√e

* Write a numeric expression such as 8x8x8x8x8 in exponential form

* The square root of n denoted as √n

* The square root of the fraction n/m denoted as √n/m

* n raised to the x

* n raised to the x

* Product of up to 5 square roots: √a√b√c√d√e

* Write a numeric expression such as 8x8x8x8x8 in exponential form

Sum of the First (n) Numbers

Free Sum of the First (n) Numbers Calculator - Determines the sum of the first (n)

* Whole Numbers

* Natural Numbers

* Even Numbers

* Odd Numbers

* Square Numbers

* Cube Numbers

* Fourth Power Numbers

* Whole Numbers

* Natural Numbers

* Even Numbers

* Odd Numbers

* Square Numbers

* Cube Numbers

* Fourth Power Numbers

Take a look at the following sums: 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 +

Take a look at the following sums:
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
a. Come up with a conjecture about the sum when you add the first *n* odd numbers. For example, when you added the first 5 odd numbers (1 + 3 + 5 + 7 + 9), what did you get? What if wanted to add the first 10 odd numbers? Or 100?
b. Can you think of a geometric interpretation of this pattern? If you start with one square and add on three more, what can you make? If you now have 4 squares and add on 5 more, what can you make?
c. Is there a similar pattern for adding the first n even numbers?
2 = 2
2 + 4 = 6
2 + 4 + 6 = 12
2 + 4 + 6 + 8 = 20
a. The formula is [B]n^2[/B].
The sum of the first 10 odd numbers is [B]100[/B] seen on our s[URL='http://www.mathcelebrity.com/sumofthefirst.php?num=10&pl=Odd+Numbers']um of the first calculator[/URL]
The sum of the first 100 odd numbers is [B]10,000[/B] seen on our [URL='http://www.mathcelebrity.com/sumofthefirst.php?num=100&pl=Odd+Numbers']sum of the first calculator[/URL]
b. Geometric is 1, 4, 9 which is our [B]n^2[/B]
c. The sum of the first n even numbers is denoted as [B]n(n + 1)[/B] seen here for the [URL='http://www.mathcelebrity.com/sumofthefirst.php?num=+10&pl=Even+Numbers']first 10 numbers[/URL]

The difference between the product of 4 and a number and the square of a number

The difference between the product of 4 and a number and the square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The product of 4 and a number:
4x
The square of a number means we raise x to the power of 2:
x^2
The difference between the product of 4 and a number and the square of a number:
[B]4x - x^2[/B]

The difference between the squares of two consecutive numbers is 141. Find the numbers

The difference between the squares of two consecutive numbers is 141. Find the numbers
Take two consecutive numbers:
n- 1 and n
Given a difference (d) between the squares of two consecutive numbers, the shortcut for this is:
2n - 1 = d
Proof of this:
n^2- (n - 1)^2 = d
n^2 - (n^2 - 2n + 1) = d
n^2 - n^2 + 2n - 1 = d
2n - 1 = d
Given d = 141, we have
2n - 1 = 141
Add 1 to each side:
2n = 142
Divide each side by 2:
2n/2 = 142/2
n = [B]71[/B]
Therefore, n - 1 = [B]70
Our two consecutive numbers are (70, 71)[/B]
Check your work
70^2 = 4900
71^2 = 5041
Difference = 5041 - 4900
Difference = 141
[MEDIA=youtube]vZJtZyYWIFQ[/MEDIA]

The difference between two positive numbers is 5 and the square of their sum is 169

The difference between two positive numbers is 5 and the square of their sum is 169.
Let the two positive numbers be a and b. We have the following equations:
[LIST=1]
[*]a - b = 5
[*](a + b)^2 = 169
[*]Rearrange (1) by adding b to each side. We have a = b + 5
[/LIST]
Now substitute (3) into (2):
(b + 5 + b)^2 = 169
(2b + 5)^2 = 169
[URL='https://www.mathcelebrity.com/community/forums/calculator-requests.7/create-thread']Run (2b + 5)^2 through our search engine[/URL], and you get:
4b^2 + 20b + 25
Set this equal to 169 above:
4b^2 + 20b + 25 = 169
[URL='https://www.mathcelebrity.com/quadratic.php?num=4b%5E2%2B20b%2B25%3D169&pl=Solve+Quadratic+Equation&hintnum=+0']Run that quadratic equation in our search engine[/URL], and you get:
b = (-9, 4)
But the problem asks for [I]positive[/I] numbers. So [B]b = 4[/B] is one of our solutions.
Substitute b = 4 into equation (1) above, and we get:
a - [I]b[/I] = 5
[URL='https://www.mathcelebrity.com/1unk.php?num=a-4%3D5&pl=Solve']a - 4 = 5[/URL]
[B]a = 9
[/B]
Therefore, we have [B](a, b) = (9, 4)[/B]

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered t

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as Benford's Law. For example, the following distribution represents the first digits in 231 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.
Digit, Probability
1, 0.301
2, 0.176
3, 0.125
4, 0.097
5, 0.079
6, 0.067
7, 0.058
8, 0.051
9, 0.046
[B][U]Fradulent Checks[/U][/B]
Digit, Frequency
1, 36
2, 32
3, 45
4, 20
5, 24
6, 36
7, 15
8, 16
9, 7
Complete parts (a) and (b).
(a) Using the level of significance α = 0.05, test whether the first digits in the allegedly fraudulent checks obey Benford's Law. Do the first digits obey the Benford's Law?

Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

The product of a number and its square is less than 8

Let the number be x.
Let the square be x^2.
So we have (x)(x^2) = x^3 < 8
Take the cube root of this, we get x = 2

The square of a number added to its reciprocal

The square of a number added to its reciprocal
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
the square of x mean we raise x to the power of 2. It's written as:
x^2
The reciprocal of x is 1/x
We add these together to get our final algebraic expression:
[B]x^2 + 1/x
[MEDIA=youtube]ZHut58-AoDU[/MEDIA][/B]

The square of a number increased by 7 is 23

The square of a number increased by 7 is 23
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
x
The square of a number means we raise x to the power of 2:
x^2
[I]Increased by[/I] means we add 7 to x^2
x^2 + 7
The word [I]is[/I] means an equation. So we set x^2 + 7 equal to 23:
[B]x^2 + 7 = 23[/B]

The square of a number is always nonnegative.

The square of a number is always nonnegative.
This is true, and here is why:
Suppose you have a positive number n.
n^2 = n * n
A positive times a positive is a positive
Suppose you have a negative number -n
(-n)^2 = -n * -n = n^2
A negative times a negative is a positive.

The square of a number is positive

The square of a number is positive
N ca be positive or negative, so test both scenarios:
Take a positive number n.
n^2 = n^2 * n^2 or Positive * Positive which is positive
Take a negative number n
(-n)^2 = -n * -n or Negative * Negative which is positive
(-n)^2 = n^2

The square of the difference of a number and 4

The square of the difference of a number and 4
A number means an arbitrary variable, let's call it x
The difference of a number and 4:
x - 4
The square of this difference:
[B](x - 4)^2[/B]

The square of the sum of twice a number x and y

The square of the sum of twice a number x and y
Take this in algebraic expression in 3 parts:
[LIST=1]
[*]Twice a number x means we multiply x by 2: 2x
[*]The sum of twice a number x and y means we add y to 2x above: 2x + y
[*]The square of the sum means we raise the sum (2x + y) to the second power below:
[/LIST]
[B](2x + y)^2[/B]

the square of the sum of two numbers

the square of the sum of two numbers
Let the first number be x. Let the second number be y.
The sum is:
x + y
Now we square that sum by raising the sum to a power of 2:
[B](x + y)^2[/B]

the square root of twice a number is 4 less than the number

Write this out, let the number be x.
sqrt(2x) = x - 4 since 4 less means subtract
Square each side:
sqrt(2x)^2 = (x - 4)^2
2x = x^2 - 8x + 16
Subtract 2x from both sides
x^2 - 10x + 16 = 0
Using the [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2+-+10x+%2B+16+%3D+0&pl=Solve+Quadratic+Equation&hintnum=0']quadratic calculator[/URL], we get two potential solutions
x = (2, 8)
Well, 2 does not work, since sqrt(2*2) = 2 which is not 4 less than 2
However, 8 does work:
sqrt(2*8) = sqrt(16) = 4, which is 4 less than the number 8.

the sum of 16 squared and a number x

the sum of 16 squared and a number x
16 squared:
16^2
The sum of this and a number x
[B]x + 16^2[/B]

The sum of 3 times the square of a number and negative 7

The sum of 3 times the square of a number and negative 7
[U]The phrase [I]a number[/I] means an arbitrary variable, let's call it x:[/U]
x
[U]The square of a number means we raise x to the power of 2:[/U]
x^2
[U]3 times the square of a number:[/U]
3x^2
[U]The sum of 3 times the square of a number and negative 7[/U]
[B]3x^2 - 7[/B]

The sum of a number and its square is 72. find the numbers?

The sum of a number and its square is 72. find the numbers?
Let the number be n. We have:
n^2 + n = 72
Subtract 72 from each side:
n^2 + n - 72 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-72%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we have:
[B]n = 8 or n = -9
[/B]
Since the numbers do not state positive or negative, these are the two solutions.

The sum Of a number squared and 14

The sum Of a number squared and 14.
A number means an arbitrary variable, let's call it x.
Squared means we raise x to the 2nd power: x^2
The sum means we add x^2 to 14 to get our algebraic expression below:
[B]x^2 + 14[/B]

The sum of the square of a number and 7 times a number

The sum of the square of a number and 7 times a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Square the number:
x^2
7 times the number means we multiply x by 7:
7x
The sum means we add x^2 and 7x
[B]x^2 + 7x[/B]

The sum of the squares of two consecutive positive integers is 61. Find these two numbers.

The sum of the squares of two consecutive positive integers is 61. Find these two numbers.
Let the 2 consecutive integers be x and x + 1. We have:
x^2 + (x + 1)^2 = 61
Simplify:
x^2 + x^2 + 2x + 1 = 61
2x^2 + 2x + 1 = 61
Subtract 61 from each side:
2x^2 + 2x - 60 = 0
Divide each side by 2
x^2 + x - 30
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-30&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation calculator[/URL], we get:
x = 5 and x = -6
The question asks for [I]positive integers[/I], so we use [B]x = 5. [/B]This means the other number is [B]6[/B].

total of a number and the square of a number

total of a number and the square of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The square of a number means we raise x to the power of 2. x^2
The total means we add x squared to x:
[B]x + x^2[/B]

twice a number subtracted from the square root of the same number

twice a number subtracted from the square root of the same number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Twice a number means we multiply x by 2:
2x
Square root of the same number:
sqrt(x)
twice a number subtracted from the square root of the same number
[B]sqrt(x) - 2x[/B]

twice the square root of a number increased by 5 is 23

twice the square root of a number increased by 5 is 23
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
The square root of a number means we raise x to the 1/2 power:
sqrt(x)
the square root of a number increased by 5 means we add 5 to sqrt(x):
sqrt(x) + 5
twice the square root of a number increased by 5 means we multiply sqrt(x) + 5 by 2:
2(sqrt(x) + 5)
The phrase [I]is 23[/I] means we set 2(sqrt(x) + 5) equal to 23:
[B]2(sqrt(x) + 5) = 23[/B]

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.
If sum of two numbers is 20, let one number be x. Then the other number would be 20 - x.
The sum of their squares is:
x^2+(20 - x)^2
Expand this and we get:
x^2 + 400 - 40x + x^2
Combine like terms:
2x^2 - 40x + 400
Rewrite this:
2(x^2 - 20x + 100 - 100) + 400
2(x - 10)^2 - 200 + 400
2(x−10)^2 + 200
The sum of squares of two numbers is sum of two positive numbers, one of which is a constant of 200.
The other number, 2(x - 10)^2, can change according to the value of x. The least value could be 0, when x=10
Therefore, the minimum value of sum of squares of two numbers is 0 + 200 = 200 when x = 10.
If x = 10, then the other number is 20 - 10 = 10.

What is the value of 998^2 – 2^2?

A) 988,036
B) 990,000
C) 995,988
D) 996,000
E) 1,000,000
This is a difference of squares.
The formula for 2 numbers a and b is:
a^2 - b^2 = (a + b)(a - b)
In our problem, we have a = 998 and b = 2:
998^2 – 2^2 = (998 + 2)(998 - 2)
998^2 – 2^2 = 1000(996)
Multiplying by 1000 means we move the decimal place of the other number 3 places to the right:
998^2 – 2^2 = [B]996,000 or Answer D
[MEDIA=youtube]IeKLs8Ds-No[/MEDIA][/B]

When 28 is subtracted from the square of a number, the result is 3 times the number. Find the negati

When 28 is subtracted from the square of a number, the result is 3 times the number. Find the negative solution.
Let the number be n.
Square of a number:
n^2
28 is subtracted from the square of a number:
n^2 - 28
3 times the number:
3n
[I]The result is[/I] mean an equation, so we set n^2 - 28 = 3n
n^2 - 28 = 3n
Subtract 3n from each side:
n^2 - 3n - 28 = 3n - 3n
The right side cancels to 0, so we have:
n^2 - 3n - 28 = 0
This is a quadratic equation in standard form, so we [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-28%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']use our quadratic calculator[/URL] to solve:
We get two solutions for n:
n = (-4, 7)
The question asks for the negative solution, so our answer is:
[B]n = -4[/B]

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positiv

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positive solution.
Let the number be n. We have:
n^2 - 4 = 3n
Subtract 3n from each side:
n^2 - 3n - 4 = 0
[URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Typing this quadratic equation into the search engine[/URL], we get:
n = (-1, 4)
The problem asks for the positive solution, so we get [B]n = 4[/B].

When 54 is subtracted from the square of a number, the result is 3 times the number.

When 54 is subtracted from the square of a number, the result is 3 times the number.
This is an algebraic expression. Let's take it in parts.
The phrase [I]a number[/I] means an arbitrary variable, let's call it "x".
x
Square the number, means raise it to the 2nd power:
x^2
Subtract 54:
x^2 - 54
The phrase [I]the result[/I] means an equation, so we set x^2 - 54 equal to 3
[B]x^2 - 54 = 3[/B]

You and your friend are playing a number-guessing game. You ask your friend to think of a positive n

You and your friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add three. If your friend's final answer is 53, what was the original number chosen?
Let n be our original number.
Square the number means we raise n to the power of 2:
n^2
Multiply the result by 2:
2n^2
And then add three:
2n^2 + 3
If the friend's final answer is 53, this means we set 2n^2 + 3 equal to 53:
2n^2 + 3 = 53
To solve for n, we subtract 3 from each side, to isolate the n term:
2n^2 + 3 - 3 = 53 - 3
Cancel the 3's on the left side, and we get:
2n^2 = 50
Divide each side of the equation by 2:
2n^2/2 = 50/2
Cancel the 2's, we get:
n^2 = 25
Take the square root of 25
n = +-sqrt(25)
n = +-5
We are told the number is positive, so we discard the negative square root and get:
n = [B]5[/B]