ACT Math : How to factor a variable

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : Factoring

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

Possible Answers:

15

45

35

20

25

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 #1 : How To Factor A Variable

Factor 12x3y+ 156x2y3

Possible Answers:

12xy(xy + 13)

x2y3(xy + 13)

12x2y3(xy + 13)

12x2y3

Correct answer:

12x2y3(xy + 13)

Explanation:

The common factors are 12, x2, and y3.

So 12x2y3(xy + 13)

Example Question #3 : Factoring

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

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

Possible Answers:

\dpi{100} \small 3,8

\dpi{100} \small -4,6

\dpi{100} \small 4,6

\dpi{100} \small 3,-8

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

Correct answer:

\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

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

Example Question #4 : Factoring

Simplify:

Possible Answers:

Correct answer:

Explanation:

 factors to

One  cancels from the bottom, leaving 

Example Question #5 : Factoring

Factor:  

Possible Answers:

Correct answer:

Explanation:

 

In the form of  you must find two numbers which add to give you  and multiply to give you  and then put them in the form of ( + number) ( + number)

Therefore  is the answer.

To check, multiply the two expressions out and it should equal 

Example Question #6 : Factoring

Factor the following expression: 

Possible Answers:

The expression is already simplified as much as possible.

Correct answer:

Explanation:

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

Remember that 

Thus:

Example Question #7 : Factoring

Factor the following expression:

Possible Answers:

Correct answer:

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,

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