ACT Math : Equations / Inequalities

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #4 : 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:

Explanation:

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 #5 : How To Find The Solution To A Quadratic Equation

If x^2=25 and y^2=81, what is the greatest value that (x-y)^2 can have?

Possible Answers:

100

16

196

25

81

Correct answer:

196

Explanation:

Solving for x yields -5 and5. Solving for y yields -9 and 9.

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

Example Question #6 : How To Find The Solution To A Quadratic Equation

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

Possible Answers:

Correct answer:

Explanation:

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 #7 : 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:

Explanation:

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

So the equation to solve becomes

 

or

x^{2} + 2x - 99 = 0

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

Example Question #8 : 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:

1

51

73

49

1

Correct answer:

51

Explanation:

We can generate an equation for the number of home runs he has hit, x : x2 - 50x = 50.  Reordering this, we get : x2 - 50x - 50 = 0.  Using the quadratic equation: x = (-b± √(b2-4ac)) / (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 #9 : How To Find The Solution To A Quadratic Equation

If (x + a)(x + b) = x2  9x + 18, what are the values for a and b?

Possible Answers:

a = 3, b = 6

a = 6, b =3

a = 3, b = 6

a = 3, b = 6

a = 3, b = 6

Correct answer:

a = 3, b = 6

Explanation:

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 #10 : How To Find The Solution To A Quadratic Equation

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

Possible Answers:

27/22

7/11

–29/22

4

19/22

Correct answer:

27/22

Explanation:

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.

Example Question #11 : How To Find The Solution To A Quadratic Equation

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

Possible Answers:

Correct answer:

Explanation:

Let  be defined as the lower number, and  as the greater number.

We know that the first number times the second is , so the equation to solve becomes .

Distributing the gives us a polynomial, which we can solve by factoring.

x^{2} + 3x - 180 = 0

and

The question tells us that the integers are positive; therefore, .

If , and the second number is , then the second number is .

The sum of these numbers is .

 

Example Question #12 : How To Find The Solution To A Quadratic Equation

Find the solutions of this quadratic equation:

4y3 - 4y2 = 8y

Possible Answers:

 –1, 2

–2, 4

1, 2

–1, –2

2, 4

Correct answer:

 –1, 2

Explanation:

4y3 - 4y2 = 8y

Divide by y and set equal to zero.

4y2 - 4y – 8 = 0

(2y + 2)(2y – 4) = 0

2y + 2 = 0

2y = –2

y = –1

2y – 4 = 0

2y = 4

y = 2

Example Question #271 : Equations / Inequalities

Which of the following is a solution to:

Possible Answers:

Correct answer:

Explanation:

You may use the quadratic formula (where ), which yields two answers,  and .

Since the only solution that appears in the answer list is , we choose .

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