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 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 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 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. 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]

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

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]

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 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 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 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]

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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 ?:[/U] ? = b2 - 4ac: ? = 62 - 4 x 2 x -108 ? = 36 - -864 ? = 900 <--- Discriminant Since ? is greater than zero, we can expect two real and unequal roots. [U]Step 3 - take the square root of the discriminant ?:[/U] ?? = ?(900) ?? = 30 [U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U] Numerator 1 = -b + ?? 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 - ?? 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

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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

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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. 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. 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 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 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]

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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 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 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]

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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 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. 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]

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]

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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

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

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

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 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 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. 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 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 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 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 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]

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 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 [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? 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. 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 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. 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 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 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 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. 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.

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 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. 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 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]