### All ACT Math Resources

## Example Questions

### Example Question #11 : Quadratic Equations

What is the sum of the two solutions of the equation x^{2} + 5x – 24 = 0?

**Possible Answers:**

-24/5

-5

3

24/5

-24

**Correct answer:**

-5

First you must find the solutions to the equation. This can be done either by using the quadratic formula or by simply finding two numbers whose sum is 5 and whose product is -24 and factoring the equation into (x + 8)(x – 3) = 0. The solutions to the equation are therefore -8 and 3, giving a sum of -5.

### Example Question #1 : How To Find The Solution To A Quadratic Equation

The height of a ball (in feet) after it is thrown in the air is given by the expression

*s*(*t*) = –*t*^{2} + 4*t*^{}

where *t* is time in seconds. The ball is thrown from ground level at *t* = 0. How many seconds will pass before the ball reaches the ground again?

**Possible Answers:**

6

8

10

4

2

**Correct answer:**

4

Notice that when the ball is at ground level, the height is zero. Setting *s *(*t*) equal to zero and solving for *t* will then give the times when the ball is at the ground.

*–t*^{2} + 4*t* =0

*t*(4 – *t*) = 0

*t* = 0, *t* = 4

The ball returns to the ground after 4 seconds.

### Example Question #1 : How To Find The Solution To A Quadratic Equation

Two positive consecutive multiples of 3 have a product of 180. What is the sum of the two numbers?

**Possible Answers:**

**Correct answer:**

Define varables as = the first number and = the second number. The product of the numbers is . Solve the resulting quadratic equation by factoring and setting each factor to zero. The numbers are 12 and 15 and the sum is 27.

### Example Question #1 : How To Find The Solution To A Quadratic Equation

Given the equation: .

What is the product of the solutions of the quadratic equation?

**Possible Answers:**

**Correct answer:**

We are initially presented with a quadratic equation, . To begin we must factor this equation.

The multiples of 15 are (15 and 1) and (3 and 5). The only multiples that add or subtract to are 3 and 5. Hence we use these as our binomial numbers. . We must now decide on the signs. Because we need to add or subtract 5 and 3 to get to , both signs must be negative: .

From this point we need to switch gears to find solutions to the equation. What numbers would make this equation equal 0?

At this point split the equation into two parts.

and and solve.

and . Both of these numbers inserted into the original equation will produce a result of 0.

Now the question itself is asking for the product of the solutions to the equation, or , which equals 15, therefore 15 is our answer.

### Example Question #1 : How To Find The Solution To A Quadratic Equation

If and , what is the greatest value that can have?

**Possible Answers:**

**Correct answer:**

Solving for yields and. Solving for yields and .

The greatest difference between these two numbers is 14, and 14 squared is 196.

### Example Question #17 : Quadratic Equations

Two positive consecutive multiples of three have a product of 108. What is their sum?

**Possible Answers:**

**Correct answer:**

Let = 1st number

and

= 2nd number

So the equation to solve becomes

or

We factor to solve the quadratic equation to get 9 and 12 and their sum is 21.

### Example Question #1 : How To Find The Solution To A Quadratic Equation

Two consecutive positive odd numbers have a product of 99, What is the sum of the two numbers?

**Possible Answers:**

**Correct answer:**

Let = 1st odd number and = 2nd odd number.

So the equation to solve becomes

or

Solving the quadratic equation by factoring gives 9 and 11, so the sum is 20

### Example Question #1 : How To Find The Solution To A Quadratic Equation

When asked how many home runs he hit in a season, Pablo Sanchez responded with, "If you square the number of home runs and subtract 50 times the number of home runs, it is equivalent to 50." How many home runs has Pablo hit?

**Possible Answers:**

73

51

1

49

*–*1

**Correct answer:**

51

We can generate an equation for the number of home runs he has hit, x : x^{2 }- 50x = 50. Reordering this, we get : x^{2 }- 50x - 50 = 0. Using the quadratic equation: x = (-b± √(b^{2}-4ac)) / (2a). In this case, a = 1, b = -50, c = -50. Plugging in these values, we obtain the simplified equation, x = (50±51.96)/2. Therefore, x = 50.98, -0.98. Because it doesn't make sense to have a negative number of home runs, x = 50.98, which rounds up to 51 home runs.

### Example Question #1 : How To Find The Solution To A Quadratic Equation

If (x + a)(x + b) = x^{2} *–* 9x + 18, what are the values for a and b?

**Possible Answers:**

a = 6, b =3

a = *–*3, b = 6

a = 3, b = *–*6

a = 3, b = 6

a = *–*3, b = *–*6

**Correct answer:**

a = *–*3, b = *–*6

a = *–*3, b = *–*6. The sum of a and b have to be equal to *–*9, and they have to multiply together to get +18. If a = *–*3 and b = *–*6, (*–*3) + (*–*6) = (*–*9) and (*–*3)(*–*6) = 18.

### Example Question #1 : How To Find The Solution To A Quadratic Equation

Which of the following is the closest approximate solution for x where 11x^{2 }– 7x – 8 = 0?

**Possible Answers:**

–29/22

19/22

7/11

4

27/22

**Correct answer:**

27/22

Apply the quadratic formula directly to get [7 ± (49 – 4 * 11 * –8)^{0.5}]/22, → [7 ± (≈ 20)]/22

So our approximate answers are 27/22 and –13/22, and 27/22 is our answer.

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