### All ACT Math Resources

## Example Questions

### Example Question #1 : Factoring

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

**Possible Answers:**

20

45

15

35

25

**Correct answer:**

35

Define the variables as x = 1st number and x + 5 = 2nd number, so the product is given as x(x + 5) = 300, which becomes x^{2} + 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 12*x*^{3}*y*^{4 }+ 156*x*^{2}*y*^{3}

**Possible Answers:**

12*x*^{2}*y*^{3}(*xy *+ 13)

12*xy*(*xy *+ 13)

*x*^{2}*y*^{3}(*xy *+ 13)

12*x*^{2}*y*^{3}

**Correct answer:**

12*x*^{2}*y*^{3}(*xy *+ 13)

The common factors are 12, x^{2}, and y^{3}.

So 12*x*^{2}*y*^{3}(*xy* + 13)

### Example Question #3 : Factoring

Solve for all solutions of :

**Possible Answers:**

**Correct answer:**

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

Thus the solutions of are 4 and 6.

### Example Question #4 : Factoring

Simplify:

**Possible Answers:**

**Correct answer:**

factors to

One cancels from the bottom, leaving

### Example Question #5 : Factoring

Factor:

**Possible Answers:**

**Correct answer:**

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:**

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:**

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,