19 results

A rational expression is undefined when what is 0?

A rational expression is undefined when what is 0?
The [B]denominator[/B]. Because division by zero is undefined.

A rational number is such that when you multiply it by 7/3 and subtract 3/2 from the product, you ge

A rational number is such that when you multiply it by 7/3 and subtract 3/2 from the product, you get 92. What is the number?
Let the rational number be x. We're given:
7x/3 - 3/2 = 92
Using a common denominator of 3*2 = 6, we rewrite this as:
14x/6 - 9/6 = 92
(14x - 9)/6 = 92
Cross multiply:
14x - 9 = 92 * 6
14x - 9 = 552
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=14x-9%3D552&pl=Solve']type this equation into our search engine [/URL]and we get:
x = [B]40.07[/B]

Cubic Equation

Free Cubic Equation Calculator - Solves for cubic equations in the form ax^{3} + bx^{2} + cx + d = 0 using the following methods:

1) Solve the long way for all 3 roots and the discriminant Δ

2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.

1) Solve the long way for all 3 roots and the discriminant Δ

2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.

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

Irrational Numbers Between

Free Irrational Numbers Between Calculator - This calculator determines all irrational numbers between two numbers

Pleasantburg has a population growth model of P(t)=at2+bt+P0 where P0 is the initial population. Sup

Pleasantburg has a population growth model of P(t)=at^2+bt+P0 where P0 is the initial population. Suppose that the future population of Pleasantburg t years after January 1, 2012, is described by the quadratic model P(t)=0.7t^2+6t+15,000. In what month and year will the population reach 19,200?
Set P(t) = 19,200
0.7t^2+6t+15,000 = 19,200
Subtract 19,200 from each side:
0.7t^2+6t+4200 = 0
The Quadratic has irrational roots. So I set up a table below to run through the values. At t = 74, we pass 19,200. Which means we add 74 years to 2012: 2012 + 74 = [B]2086[/B]
t 0.7t^2 6t Add 15000 Total
1 0.7 6 15000 15006.7
2 2.8 12 15000 15014.8
3 6.3 18 15000 15024.3
4 11.2 24 15000 15035.2
5 17.5 30 15000 15047.5
6 25.2 36 15000 15061.2
7 34.3 42 15000 15076.3
8 44.8 48 15000 15092.8
9 56.7 54 15000 15110.7
10 70 60 15000 15130
11 84.7 66 15000 15150.7
12 100.8 72 15000 15172.8
13 118.3 78 15000 15196.3
14 137.2 84 15000 15221.2
15 157.5 90 15000 15247.5
16 179.2 96 15000 15275.2
17 202.3 102 15000 15304.3
18 226.8 108 15000 15334.8
19 252.7 114 15000 15366.7
20 280 120 15000 15400
21 308.7 126 15000 15434.7
22 338.8 132 15000 15470.8
23 370.3 138 15000 15508.3
24 403.2 144 15000 15547.2
25 437.5 150 15000 15587.5
26 473.2 156 15000 15629.2
27 510.3 162 15000 15672.3
28 548.8 168 15000 15716.8
29 588.7 174 15000 15762.7
30 630 180 15000 15810
31 672.7 186 15000 15858.7
32 716.8 192 15000 15908.8
33 762.3 198 15000 15960.3
34 809.2 204 15000 16013.2
35 857.5 210 15000 16067.5
36 907.2 216 15000 16123.2
37 958.3 222 15000 16180.3
38 1010.8 228 15000 16238.8
39 1064.7 234 15000 16298.7
40 1120 240 15000 16360
41 1176.7 246 15000 16422.7
42 1234.8 252 15000 16486.8
43 1294.3 258 15000 16552.3
44 1355.2 264 15000 16619.2
45 1417.5 270 15000 16687.5
46 1481.2 276 15000 16757.2
47 1546.3 282 15000 16828.3
48 1612.8 288 15000 16900.8
49 1680.7 294 15000 16974.7
50 1750 300 15000 17050
51 1820.7 306 15000 17126.7
52 1892.8 312 15000 17204.8
53 1966.3 318 15000 17284.3
54 2041.2 324 15000 17365.2
55 2117.5 330 15000 17447.5
56 2195.2 336 15000 17531.2
57 2274.3 342 15000 17616.3
58 2354.8 348 15000 17702.8
59 2436.7 354 15000 17790.7
60 2520 360 15000 17880
61 2604.7 366 15000 17970.7
62 2690.8 372 15000 18062.8
63 2778.3 378 15000 18156.3
64 2867.2 384 15000 18251.2
65 2957.5 390 15000 18347.5
66 3049.2 396 15000 18445.2
67 3142.3 402 15000 18544.3
68 3236.8 408 15000 18644.8
69 3332.7 414 15000 18746.7
70 3430 420 15000 18850
71 3528.7 426 15000 18954.7
72 3628.8 432 15000 19060.8
73 3730.3 438 15000 19168.3
74 3833.2 444 15000 19277.2

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]

Prove the sum of any two rational numbers is rational

Take two integers, r and s.
We can write r as a/b for integers a and b since a rational number can be written as a quotient of integers
We can write s as c/d for integers c and d since a rational number can be written as a quotient of integers
Add r and s:
r + s = a/b + c/d
With a common denominator bd, we have:
r + s = (ad + bc)/bd
Because a, b, c, and d are integers, ad + bc is an integer since rational numbers are closed under addition and multiplication.
Since b and d are non-zero integers, bd is a non-zero integer.
Since we have the quotient of 2 integers, r + s is a rational number.
[MEDIA=youtube]0ugZSICt_bQ[/MEDIA]

Quadratic Equations and Inequalities

Free Quadratic Equations and Inequalities Calculator - Solves for quadratic equations in the form ax^{2} + bx + c = 0. Also generates practice problems as well as hints for each problem.

* Solve using the quadratic formula and the discriminant Δ

* Complete the Square for the Quadratic

* Factor the Quadratic

* Y-Intercept

* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)^{2} + k

* Concavity of the parabola formed by the quadratic

* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.

* Solve using the quadratic formula and the discriminant Δ

* Complete the Square for the Quadratic

* Factor the Quadratic

* Y-Intercept

* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)

* Concavity of the parabola formed by the quadratic

* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.

Quartic Equations

Free Quartic Equations Calculator - Solves quartic equations in the form ax^{4} + bx^{3} + cx^{2} + dx + e using the following methods:

1) Solve the long way for all roots and the discriminant Δ

2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.

1) Solve the long way for all roots and the discriminant Δ

2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.

Rational Exponents - Fractional Indices

Free Rational Exponents - Fractional Indices Calculator - This calculator evaluates and simplifies a rational exponent expression in the form a^{b/c} where a is any integer *or* any variable [a-z] while b and c are integers. Also evaluates the product of rational exponents

Rational Number Subtraction

Free Rational Number Subtraction Calculator - Subtracting 2 numbers, this shows an equivalent operations is adding the additive inverse. p - q = p + (-q)

Rational Numbers

Free Rational Numbers Calculator - This lesson walks you through what rational numbers are, how to write rational numbers, rational number notation, and what's included in rational numbers

Rational Numbers Between

Free Rational Numbers Between Calculator - This calculator determines all rational numbers between two numbers

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

Sean is helping his dad build a tiled walkway in their backyard. The walkway will be 606060 feet lon

Sean is helping his dad build a tiled walkway in their backyard. The walkway will be 60 feet long and 2 feet wide. The local hardware store sells tiles which are 2 by 2 feet and come in boxes of 6.
There isn't a calculator for Rational Word Problems.

Simplify 2/sqrt(5)

To do this, we need rationalize the denominator. This means get rid of the radical:
Multiply top and bottom by sqrt(5)
2sqrt(5)/sqrt(5) * sqrt(5)
sqrt(5) * sqrt(5) = sqrt(25) so we have:
2sqrt(5)/sqrt(25)
sqrt(25) = 5, so we have:
[B]2sqrt(5)/5[/B]
[MEDIA=youtube]jearVN9LhBE[/MEDIA]

Synthetic Division

Free Synthetic Division Calculator - Using Ruffinis Rule, this performs synthetic division by dividing a polynomial with a maximum degree of 6 by a term (x ± c) where c is a constant root using the factor theorem. The calculator returns a quotient answer that includes a remainder if applicable. Also known as the Rational Zero Theorem

X is such that X belongs to rational numbers and X is less than or equal to 1 and greater than 0

X is such that X belongs to rational numbers and X is less than or equal to 1 and greater than 0
Greater than 0 means we don't include 0
0 <
less than or equal to 1 means we include 1:
[B]0 < x <= 1[/B]