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ACT Math : Variables

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Example Questions

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Example Question #1 : Binomial Denominators

For the equation $x^{2}-10x+21=0$ , what is(are) the solution(s) for ?

Possible Answers:

2,5

3,7

-3,-7

Correct answer:

3,7

Explanation:

$x^{2}-10x+21=0$ , can be factored to (x -7)(x-3) = 0. Therefore, x-7 = 0 and x-3 = 0. Solving for x in both cases, gives 7 and 3.

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Example Question #2 : How To Find The Solution Of A Rational Equation With A Binomial Denominator

Simplify:

$\frac{x^2+3x+4x+12}{x+4}$

Possible Answers:

x+3

$\frac{x^2+7x+12}{x+4}$

x-3

x+4

x-4

Correct answer:

x+3

Explanation:

In order to begin this kind of a problem, it's key to look at parts of the rational expression that can be simplified.

In this case, the denominator is an already-simplified binomial; however, the numerator can be factored through "factoring by grouping." This can be a helpful idea to keep in mind when you come across a polynomial with four terms and simplifying is involved.

${x^2+3x+4x+12}$ can be simplified first by removing the common factor of from the first two terms and the common factor of from the last two terms:

x(x+3)+4(x+3)

This leaves two terms that are identical (x+3) and their coefficients, which can be combined into another term to complete the factoring:

(x+4)(x+3)

Consider the denominator; the quantity (x+4) appears, so the (x+4) in the numerator and in the denominator can be cancelled out. The simplified expression is then left as (x+3) .

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Example Question #3 : How To Find The Solution Of A Rational Equation With A Binomial Denominator

Simplify:

$\frac{x^2+9x-10}{x-1}$

Possible Answers:

(x-10)

(x+2)

(x+1)

(x+10)

(x-1)

Correct answer:

(x+10)

Explanation:

In order to begin this kind of a problem, it's key to look at parts of the rational expression that can be simplified.

In this case, the denominator is an already-simplified binomial; however, the numerator can be factored.

x^2+9x-10

The roots will be numbers that sum up to but have the product of .

The options include:

(-1)(10) = -10
(1)(-10) = -10
(2)(-5) = -10
(-2)(5)= -10

When these options are summed up:

-1+10=9
1-10=-9
2-5=-3
-2+5=3

We can negate the last three options because the first option of and fulfill the requirements. Therefore, the numerator can be factored into the following:

${x^2+9-10}=(x-1)(x+10)$

Because the quantity (x-1) appears in the denominator, this can be "canceled out." This leaves the final answer to be the quantity (x+10) .

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Example Question #1 : How To Simplify Binomials

Solve for x when 6x – 4 = 2x + 5

Possible Answers:

1/4

9/4

Correct answer:

9/4

Explanation:

Solve by simplifying:

6x – 4 = 2x + 5

6x = 2x + 9

4x = 9

x = 9/4

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Example Question #31 : Polynomials

What is the value of the following equation if t = 3 and v = –4 ?

$\small 2t^2+4vt+2v^2$

Possible Answers:

Correct answer:

Explanation:

Substitute the numbers 3 and –4 for t and v, respectively.

$\small 2(3)^2+4(-4)(3)+2(-4)^2$

$\small 18+(-48)+32=2$

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Example Question #1 : How To Simplify Binomials

Simplify the following binomial:

$4x^2 + 16x - 4 - 2x^{2}$

Possible Answers:

x^2 - 4x +4

x^2 + 8x - 2

x^2 + 4x - 4

x^2 + 8x -4

2x^2 - 4x +16

Correct answer:

x^2 + 8x - 2

Explanation:

The equation that is presented is:

$4x^2 + 16x - 4 - 2x^{2}$

To get the correct answer, you first need to combine all of the like terms. So, you can subtract the 2x^2 from the 4x^2 , leaving you with:

2x^2 + 16x - 4

From there, you can reduce the numbers by their greatest common denominator, in this case, :

x^2 + 8x - 2

Then you have arrived at your final answer.

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Example Question #4 : How To Simplify Binomials

Simplify the following binomial expression:

16x^2 - 14x + 12x - 8 + 2

Possible Answers:

8x^2 + 2x -3

x^2 - 8x - 3

4x^2 - 4x - 4

8x^2 - x - 3

2x^2 - 4x - 3

Correct answer:

8x^2 - x - 3

Explanation:

First, combine all of the like terms that you are able:

16x^2 - 2x - 6

Then, reduce by the greatest common denominator (in this case, ):

8x^2 - x - 3

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Example Question #5 : How To Simplify Binomials

Simplify the following binomial:

$17x^{2} - 15x + 3x^{2 }+10x$

Possible Answers:

5x(4x-1)

5x^2(4x-x)

10x^2 - x

5(4x-x)

5x^2 - 3x

Correct answer:

5x(4x-1)

Explanation:

The equation presented in the problem is:

$17x^{2} - 15x + 3x^{2 }+10x$

First you have to combine the like terms, i.e. combining all instances of x^2 and :

20x^2 - 5x

Then, you can factor out the common to get your answer

5x(4x-1)

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Example Question #21 : Binomials

Which of the following expressions is equivalent to: 6x (m²+yx²–3)?

Possible Answers:

6xm² + 7x³ -18

6xm² + 6yx³ -18

6xm² + 6yx² -18x

6xm² + 6yx³ -18x

xm² + 7x³ -18

Correct answer:

6xm² + 6yx³ -18x

Explanation:

6x (m²+yx²–3)= 6x∙m²+ 6xyx² – 6x∙3= 6xm² + 6yx³-18x (Use Distributive Property)

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Example Question #1 : Binomials And Foil

Which of the following expressions is equivalent to: $7x(x^{2}+xz^{2}-7)$ ?

Possible Answers:

$x^{3}+x^{2}z^{2}-49x$

$7x^{3}+7z^{2}-49x$

$7x^{3}+7x^{2}z^{2}-49$

$7x^{2}+x^{2}z^{2}-49x$

$7x^{3}+7x^{2}z^{2}-49x$

Correct answer:

$7x^{3}+7x^{2}z^{2}-49x$

Explanation:

Use the distributive property to multiply by all of the terms in $(x^{2}+xz^{2}-7)$ :

$(7x\cdot x^{2})+(7x\cdot xz^{2}) + (7x\cdot -7)=$

$7x^{3}+7x^{2}z^{2}-49x$

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ACT Math : Variables

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Binomial Denominators

Example Question #2 : How To Find The Solution Of A Rational Equation With A Binomial Denominator

Example Question #3 : How To Find The Solution Of A Rational Equation With A Binomial Denominator

Example Question #1 : How To Simplify Binomials

Example Question #31 : Polynomials

Example Question #1 : How To Simplify Binomials

Example Question #4 : How To Simplify Binomials

Example Question #5 : How To Simplify Binomials

Example Question #21 : Binomials

Example Question #1 : Binomials And Foil

All ACT Math Resources

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