ACT Math : How to factor a variable

Study concepts, example questions & explanations for ACT Math

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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:

25

15

35

45

20

Correct answer:

35

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.

Example Question #2 : Factoring

Factor 12x3y+ 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)

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. 

Example Question #2 : Factoring

Simplify:

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

Possible Answers:

\(\displaystyle x-3\)

\(\displaystyle x^2+6x+9\)

\(\displaystyle x^2+5x+6\)

\(\displaystyle x+3\)

Correct answer:

\(\displaystyle x+3\)

Explanation:

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

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

Example Question #3 : Factoring

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

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:

\(\displaystyle x^{2}+2x-24\) 

In the form of \(\displaystyle ax^{2}+bx+c\) 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)

\(\displaystyle 6\ast-4=24\)

\(\displaystyle 6+-4=2\)

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

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

Example Question #106 : Variables

Factor the following expression: 

\(\displaystyle 3x^2 +12x\)

Possible Answers:

The expression is already simplified as much as possible.

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

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

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

\(\displaystyle x(3x+12)\)

Correct answer:

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

Explanation:

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

Remember that 

\(\displaystyle x^2 = x\cdot x\)

Thus:

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

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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