### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Factor An Equation

Solve 8x^{2} – 2x – 15 = 0

**Possible Answers:**

x = -3/2 or 5/4

x = 3/2 or -5/4

x = 3/2 or 5/4

x = -3/2 or -5/4

**Correct answer:**

x = 3/2 or -5/4

The equation is in standard form, so a = 8, b = -2, and c = -15. We are looking for two factors that multiply to ac or -120 and add to b or -2. The two factors are -12 and 10.

So you get (2x -3)(4x +5) = 0. Set each factor equal to zero and solve.

### Example Question #2 : How To Factor An Equation

What is the value of m if m^{2} + 10m - 24 = 0 and m < 0?

**Possible Answers:**

-8

-2

-6

-12

-4

**Correct answer:**

-12

m^{2} + 10m - 24 = 0 is a quadratic equation, so we can use the FOIL method to find the roots of the equation.

am^{2 }+ bm - c = 0

Since a = 1, we know that we will have (m + )(m - ) = 0. We also know that one of those signs has to be negative, because our two last terms multiply to equal c and c is negative here.

We need two numbers that multiply together to equal -24, and that add up to equal 10.

The factors of 24 are:

24 and 1; 12 and 2; 8 and 3; and 6 and 4.

Since one of those factors will be negative in our factored equation, we need to find the two factors where the difference is 10.

12 - 2 = 10. This means that our two FOIL last numbers are 12 and -2.

(m + 12)(m - 2) = 0.

By the zero multiplication rule, either portion of that equation can equal 0 for the result to be 0. Thus, we have:

m + 12 = 0 (Subtract 12 from both sides.)

m = -12

and

m - 2 = 0 (Add 2 to both sides.)

m = 2

Since we want a negative answer for m, m = -12.

### Example Question #3 : How To Factor An Equation

If (x^{2 }+ 2) / 2 = (x^{2} - 6x - 1) / 5, then what is the value of x?

**Possible Answers:**

4

2

-3

3

-2

**Correct answer:**

-2

(x^{2 }+ 2) / 2 = (x^{2} - 6x - 1) / 5. We first cross-multiply to get rid of the denominators on both sides.

5(x^{2} + 2) = 2(x^{2} - 6x - 1)

5x^{2} + 10 = 2x^{2} - 12x - 2 (Subtract 2x^{2}, and add 12x and 2 to both sides.)

3x^{2} + 12x + 12 = 0 (Factor out 3 from the left side of the equation.)

3(x^{2} + 4x + 4) = 0 (Factor the equation, knowing that 2 + 2 = 4 and 2*2 = 4.)

3(x + 2)(x + 2) = 0

x + 2 = 0

x = -2

### Example Question #4 : How To Factor An Equation

Which of the following is a factor of the polynomial *x*^{2} – 6*x* + 5?

**Possible Answers:**

*x* – 6

*x* – 8

*x *+ 1

*x* + 2

*x* – 5

**Correct answer:**

*x* – 5

Factor the polynomial by choosing values that when FOIL'ed will add to equal the middle coefficient, 3, and multiply to equal the constant, 1.

*x*^{2} – 6*x *+ 5 = (*x* – 1)(*x* – 5)

Because only (*x* – 5) is one of the choices listed, we choose it.

### Example Question #5 : How To Factor An Equation

7 times a number is 30 less than that same number squared. What is one possible value of the number?

**Possible Answers:**

**Correct answer:**

Either:

or:

### Example Question #6 : How To Factor An Equation

Which of the following is equivalent to ?

**Possible Answers:**

**Correct answer:**

The answer is ^{.}

To determine the answer, must be distrbuted,

. After multiplying the terms, the expression simplifies to .

### Example Question #7 : How To Factor An Equation

For what value of b is the equation b^{2} + 6b + 9 = 0 true?

**Possible Answers:**

0

**–**3

3

5

**Correct answer:**

**–**3

Factoring leads to (b+3)(b+3)=0. Therefore, solving for b leads to -3.

### Example Question #8 : How To Factor An Equation

What is the solution to:

**Possible Answers:**

0

4

1

6

2

**Correct answer:**

4

First you want to factor the numerator from x^{2 }– 6x + 8 to (x – 4)(x – 2)

Input the denominator (x – 4)(x – 2)/(x – 2) = (x – 4) = 0, so x = 4.

### Example Question #9 : How To Factor An Equation

What is the value of where:

**Possible Answers:**

**Correct answer:**

The question asks us to find the value of , because it is in a closed equation, we can simply put all of the whole numbers on one side of the equation, and all of the containing numbers on the other side.

We utilize opposite operations to both sides by adding to each side of the equation and get

Next, we subtract from both sides, yielding

Then we divide both sides by to get rid of that on

### Example Question #10 : How To Factor An Equation

Factor the following equation:

**Possible Answers:**

**Correct answer:**

First we factor out an x then we can factor the

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