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ACT Math : How to factor a variable

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Example Questions

Example Question #1 : How To Factor A Variable

Two consecutive positive multiples of five have a product of 300. What is their sum?

Possible Answers:

Correct answer:

Explanation:

Define the variables as x = 1st number and x + 5 = 2nd number, so the product is given as x(x + 5) = 300, which becomes x² + 5x – 300 = 0.

Factoring results in (x + 20)(x – 15) = 0, so the positive answer is 15, making the second number 20.

The sum of the two numbers is 35.

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Example Question #2 : Factoring

Factor 12x³y⁴+ 156x²y³

Possible Answers:

12x²y³

x²y³(xy + 13)

12x²y³(xy + 13)

12xy(xy + 13)

Correct answer:

12x²y³(xy + 13)

Explanation:

The common factors are 12, x², and y³.

So 12x²y³(xy + 13)

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Example Question #2 : How To Factor A Variable

Solve for all solutions of $\dpi{100} \small x$ $\displaystyle \dpi{100} \small x$:

$\dpi{100} \small 2x^{2}-10x=x^{2}-24$ $\displaystyle \dpi{100} \small 2x^{2}-10x=x^{2}-24$

Possible Answers:

$\dpi{100} \small 4,6$ $\displaystyle \dpi{100} \small 4,6$

$\dpi{100} \small -4,-6$ $\displaystyle \dpi{100} \small -4,-6$

$\dpi{100} \small 3,8$ $\displaystyle \dpi{100} \small 3,8$

$\dpi{100} \small 3,-8$ $\displaystyle \dpi{100} \small 3,-8$

$\dpi{100} \small -4,6$ $\displaystyle \dpi{100} \small -4,6$

Correct answer:

$\dpi{100} \small 4,6$ $\displaystyle \dpi{100} \small 4,6$

Explanation:

First move all of the variables to the left side of the equation. Combine similar terms, and set the equation equal to zero. Then factor the equation to get $\dpi{100} \small (x-4)(x-6)=0$ $\displaystyle \dpi{100} \small (x-4)(x-6)=0$

Thus the solutions of $\dpi{100} \small x$ $\displaystyle \dpi{100} \small x$ are 4 and 6.

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Example Question #2 : Factoring

Simplify:

$\frac{x^2+6x+9}{x+3}$ $\displaystyle \frac{x^2+6x+9}{x+3}$

Possible Answers:

x-3 $\displaystyle x-3$

$\displaystyle x^2+6x+9$

$\displaystyle x^2+5x+6$

x+3 $\displaystyle x+3$

Correct answer:

x+3 $\displaystyle x+3$

Explanation:

$x^{2}+6x+9$ $\displaystyle x^{2}+6x+9$ factors to $\displaystyle (x+3)(x+3)$

One (x+3) $\displaystyle (x+3)$ cancels from the bottom, leaving (x+3) $\displaystyle (x+3)$

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Example Question #3 : Factoring

Factor: $x^{2}+2x-24$ $\displaystyle x^{2}+2x-24$

Possible Answers:

$\displaystyle (x+6)(x+4)$

$\displaystyle (x+6)(x-4)$

$\displaystyle (x+1)(x-3)$

$\displaystyle (x+4)(x-6)$

$\displaystyle (2x+4)(x-6)$

Correct answer:

$\displaystyle (x+6)(x-4)$

Explanation:

$x^{2}+2x-24$ $\displaystyle x^{2}+2x-24$

In the form of $ax^{2}+bx+c$ $\displaystyle ax^{2}+bx+c$ you must find two numbers which add to give you $\displaystyle b$ and multiply to give you $\displaystyle c$ and then put them in the form of ($\displaystyle x$ + number) ($\displaystyle x$ + number)

$6\ast-4=24$ $\displaystyle 6\ast-4=24$

6+-4=2 $\displaystyle 6+-4=2$

Therefore $\displaystyle (x+6)(x-4)$ is the answer.

To check, multiply the two expressions out and it should equal $x^{2}+2x-24$ $\displaystyle x^{2}+2x-24$

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Example Question #106 : Variables

Factor the following expression:

$\displaystyle 3x^2 +12x$

Possible Answers:

The expression is already simplified as much as possible.

$\displaystyle 3x^2(1+4)$

3x(x+4) $\displaystyle 3x(x+4)$

$\displaystyle 3(x^2+4x)$

$\displaystyle x(3x+12)$

Correct answer:

3x(x+4) $\displaystyle 3x(x+4)$

Explanation:

To factor an expression we look for the greatest common factor.

Remember that

$x^2 = x\cdot x$ $\displaystyle x^2 = x\cdot x$

Thus:

$\\ 3x^2+12x \\ \\= 3\cdot x\cdot x+2\cdot 2\cdot 3\cdot x\\ \\=3x(x+2\cdot 2)\\ \\=3x(x+4)$ $\displaystyle \\ 3x^2+12x \\ \\= 3\cdot x\cdot x+2\cdot 2\cdot 3\cdot x\\ \\=3x(x+2\cdot 2)\\ \\=3x(x+4)$

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Example Question #6 : Factoring

Factor the following expression:

$\displaystyle x^2-13x+40$

Possible Answers:

$\displaystyle (x-10)(x-4)$

$\displaystyle (x-8)(x-5)$

$\displaystyle (x+8)(x+5)$

$\displaystyle (x+10)(x+4)$

Correct answer:

$\displaystyle (x-8)(x-5)$

Explanation:

To factor, you are looking for two factors of 40 that add to equal 13.

Factors of 40 include: (1, 40), (2, 20), (4, 10), (5, 8). Of these factors which two will add up to 13?

Also, since the first sign (-) and the second sign is (+) this tells us both binomials will be negative. This is because two negatives multiplied together will result in the positive third term, while two negatives added together will result in a larger negative number.

Thus,

$\displaystyle (x-8)(x-5)$

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ACT Math : How to factor a variable

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Factor A Variable

Example Question #2 : Factoring

Example Question #2 : How To Factor A Variable

Example Question #2 : Factoring

Example Question #3 : Factoring

Example Question #106 : Variables

Example Question #6 : Factoring

All ACT Math Resources

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