All ACT Math Resources
Example Questions
Example Question #9 : How To Find The Solution To A Quadratic Equation A1
Which of the following is the closest approximate solution for x where 11x2 – 7x – 8 = 0?
7/11
–29/22
4
27/22
19/22
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.
Example Question #21 : Quadratic Equations
Two consecutive positive multiples of three have a product of . What is the sum of the two numbers?
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.
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 A1
Find the solutions of this quadratic equation:
4y3 - 4y2 = 8y
–2, 4
–1, 2
–1, –2
2, 4
1, 2
–1, 2
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 #22 : Quadratic Equations
Which of the following is a solution to:
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 .
Example Question #23 : Quadratic Equations
2x + y3 + xy2 + y = x
If y = 1, what is x?
1
2
0
3
–1
–1
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
Example Question #24 : Quadratic Equations
What are the -intercept(s) of the following quadratic function?
and
and
and
and
None of the other answers
and
-intercepts will occur when . This yields the equation
We need to use the quadratic formula where , and .
Plugging in our values:
Simplifying:
Simplifying:
Simplifying:
Finally:
Example Question #3 : 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 , what is the width of that piece of land?
The area of a rectangle is the product of its length by its width, which we know to be equal to in our problem. We also know that the length is equal to , where represents the width of the land. Therefore, we can write the following equation:
Distributing the outside the parentheses, we get:
Subtracting from each side of the equation, we get:
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.
We can first calculate the discriminant (i.e. the part under the square root)
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):
Breaking down the square root:
We can pull two of the twos out of the square root and place a outside of it:
We can then multiply the and the :
At this point, we can reduce the equations, since each of the component parts of their right sides has a factor of :
Since width is a positive value, the answer is:
The width of the piece of land is approximately .
Example Question #15 : How To Find The Solution To A Quadratic Equation A1
Solve for x: x2 + 4x = 5
None of the other answers
-5
-5 or 1
-1 or 5
-1
-5 or 1
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
Example Question #561 : Algebra
Solve for x: (x2 – x) / (x – 1) = 1
x = 1
No solution
x = -2
x = -1
x = 2
No solution
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.
Example Question #562 : Algebra
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?
Set up two equations from the given information:
and
Substitute into the second equation:
Multiply through by .
Then divide by the coefficient of 2 to simplify your work:
Then since you have a quadratic setup, move the term to the other side (via subtraction from both sides) to set everything equal to 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:
This makes the possible solutions 10 and 12. Since 12 does not appear in the choices, is the only possible correct answer.