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ACT Math : Quadratic Equations

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Example Question #2042 : Act Math

Which of the following is the closest approximate solution for x where 11x²– 7x – 8 = 0?

Possible Answers:

27/22

–29/22

19/22

7/11

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.

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Example Question #21 : Quadratic Equations

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

Possible Answers:

Correct answer:

Explanation:

Let be defined as the lower number, and x + 3 as the greater number.

We know that the first number times the second is 180 , so the equation to solve becomes x(x + 3) = 180 .

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

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

-12*15=-180 and -12+15=3

(x-12)(x+15)=0

x=12, -15

The question tells us that the integers are positive; therefore, x=12 .

If x=12 , and the second number is x + 3 , then the second number is .

The sum of these numbers is 12+15=27 .

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Example Question #2041 : Act Math

Find the solutions of this quadratic equation:

4y³ - 4y² = 8y

Possible Answers:

1, 2

–1, –2

2, 4

–2, 4

–1, 2

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

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Example Question #21 : Quadratic Equations

Which of the following is a solution to:

9x^2-18x-27=0

Possible Answers:

Correct answer:

Explanation:

You may use the quadratic formula (where a=9, b=-18, c=-27 ), which yields two answers, and .

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

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Example Question #2042 : Act Math

2x + y³+ xy²+ y = x

If y = 1, what is x?

Possible Answers:

–1

Correct answer:

–1

Explanation:

Plug in y = 1. Then solve for x.

2x + y³+ xy²+ y = x

2x + 1 + x + 1 = x

3x + 2 = x

2x = -2

x = -1

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Example Question #25 : Quadratic Equations

What are the -intercept(s) of the following quadratic function?

y = x^2 -4x +2

Possible Answers:

None of the other answers

x = 2 and x = -2

$x = 3 + \sqrt{3}$ and $x = 3 - \sqrt{3}$

$x = 2 + \sqrt{3}$ and $x = 2 - \sqrt{3}$

$x = 2 + \sqrt{2}$ and $x = 2 - \sqrt{2}$

Correct answer:

$x = 2 + \sqrt{2}$ and $x = 2 - \sqrt{2}$

Explanation:

-intercepts will occur when y=0 . This yields the equation

0 = x^2 -4x +2

We need to use the quadratic formula where a=1 , b=-4 and c=2 .

$\frac{-b\pm(\sqrt{b^2-4ac})}{2a}$

Plugging in our values:

$\frac{4\pm\sqrt{16 - (4*1*2)}}{2}$

Simplifying:

$\frac{4\pm \sqrt{8}}{2}$

Simplifying:

$\frac{4}{2} + \frac{\sqrt{8}}{2}$

Simplifying:

$2\pm \frac{2\sqrt{2}}{2}$

Finally:

$x = 2\pm \sqrt{2}$

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Example Question #6 : How To Use The Quadratic Function

The length of a rectangular piece of land is two feet more than three times its width. If the area of the land is $125\:ft^2$ , what is the width of that piece of land?

Possible Answers:

$\frac{1-2\sqrt{94}}{3}\:ft^2$

$\frac{2-4\sqrt{94}}{6}\:ft^2$

$\frac{-1+2\sqrt{94}}{3}\:ft^2$

$\frac{-1+4\sqrt{94}}{3}\:ft^2$

$\frac{-1-2\sqrt{94}}{3}\:ft^2$

Correct answer:

$\frac{-1+2\sqrt{94}}{3}\:ft^2$

Explanation:

The area of a rectangle is the product of its length by its width, which we know to be equal to $125\:ft^2$ in our problem. We also know that the length is equal to 2+3w , where represents the width of the land. Therefore, we can write the following equation:

w(2+3w)=125

Distributing the outside the parentheses, we get:

$3w^{2}+2w=125$

Subtracting 125 from each side of the equation, we get:

$3w^{2}+2w-125=0$

We get a quadratic equation, and since there is no factor of and that adds up to , we use the quadratic formula to solve this equation.

a=3 b=2 c=-125

$w=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

We can first calculate the discriminant (i.e. the part under the square root)

$b^{2}-4ac= 2^{2}-4\times3 \times(-125)$

$4-4\times3\times(-125)$

$4-12\times(-125)$

4-(-1500)=1504

We replace that value in the quadratic formula, solving both the positive version of the formula (on the left) and the negative version of the formula (on the right):

$w=\frac{-2+\sqrt{1504}}{6}$ $w=\frac{-2+\sqrt{1504}}{6}$

Breaking down the square root:

$w=\frac{-2+\sqrt{2\times2\times2\times2\times2\times 47}}{6}$ $w=\frac{-2-\sqrt{2\times2\times2\times2\times2\times 47}}{6}$

We can pull two of the twos out of the square root and place a outside of it:

$w=\frac{-2+4\sqrt{2\times 47}}{6}$ $w=\frac{-2-4\sqrt{2\times 47}}{6}$

We can then multiply the and the :

$w=\frac{-2+4\sqrt{94}}{6}$ $w=\frac{-2-4\sqrt{94}}{6}$

At this point, we can reduce the equations, since each of the component parts of their right sides has a factor of :

$w=\frac{-1+2\sqrt{94}}{3}$ $w=\frac{-1+2\sqrt{94}}{3}$

Since width is a positive value, the answer is:

$w=\frac{-1+2\sqrt{94}}{3}$

$w=\frac{-1+2\times9.695}{3}$

$w=\frac{-1+19.39}{3}$

$w=\frac{18.39}{3}=6.13$

The width of the piece of land is approximately $6.13\:ft^2$ .

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

Solve for x: x² + 4x = 5

Possible Answers:

-1

None of the other answers

-5

-1 or 5

-5 or 1

Correct answer:

-5 or 1

Explanation:

Solve by factoring. First get everything into the form Ax² + Bx + C = 0:

x² + 4x - 5 = 0

Then factor: (x + 5) (x - 1) = 0

Solve each multiple separately for 0:

X + 5 = 0; x = -5

x - 1 = 0; x = 1

Therefore, x is either -5 or 1

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Example Question #1 : Quadratic Equation

Solve for x: (x² – x) / (x – 1) = 1

Possible Answers:

x = -1

No solution

x = -2

x = 2

x = 1

Correct answer:

No solution

Explanation:

Begin by multiplying both sides by (x – 1):

x² – x = x – 1

Solve as a quadratic equation: x² – 2x + 1 = 0

Factor the left: (x – 1)(x – 1) = 0

Therefore, x = 1.

However, notice that in the original equation, a value of 1 for x would place a 0 in the denominator. Therefore, there is no solution.

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Example Question #25 : How To Find The Solution To A Quadratic Equation

A farmer has 44 feet of fence, and wants to fence in his sheep. He wants to build a rectangular pen with an area of 120 square feet. Which of the following is a possible dimension for a side of the fence?

Possible Answers:

24 ft

$10\ ft$

$20\ ft$

$5\ ft$

$7\ ft$

Correct answer:

$10\ ft$

Explanation:

Set up two equations from the given information:

$\dpi{100} \small 120=xy$ and 44=2x+2y

Substitute $\dpi{100} \small y=\frac{120}{x}$ into the second equation:

$44=2x+2(\frac{120}{x})$

Multiply through by $\dpi{100} \small x$ .

$44x=2x^{2}+240$

Then divide by the coefficient of 2 to simplify your work:

$22x=x^{2}+120$

Then since you have a quadratic setup, move the 22x term to the other side (via subtraction from both sides) to set everything equal to 0:

$x^{2}-22x+120=0$

As you look for numbers that multiply to positive 120 and add to -22 so you can factor the quadratic, you might recognize that -12 and -10 fit the bill. This makes your factorization:

(x-10)(x-12)=0

This makes the possible solutions 10 and 12. Since 12 does not appear in the choices, $\dpi{100} \small 10\ feet$ is the only possible correct answer.

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ACT Math : Quadratic Equations

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #2042 : Act Math

Example Question #21 : Quadratic Equations

Example Question #2041 : Act Math

Example Question #21 : Quadratic Equations

Example Question #2042 : Act Math

Example Question #25 : Quadratic Equations

Example Question #6 : How To Use The Quadratic Function

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

Example Question #1 : Quadratic Equation

Example Question #25 : How To Find The Solution To A Quadratic Equation

All ACT Math Resources

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