1. Solution

Definition: The solution is the value of x that makes the function true.

What is the solution of this quadratic function?

0 = x²

1. Solve for x by taking the square root of 0 and x².

Square root of 0 = 0
Square root of x²= x
0 = x

The solution is 0.

2. Plug in 0 for x to verify the answer.

0 = x²
0 = 0²
0 = 0

2. Solution Set

Definition: The solution set includes 2 values of the input that make the function true.

What is the solution set of the following function?

k² + 11k = -28

1. Set the function equal to 0.

k² + 11k + 28 = -28 +28
k² + 11k + 28 = 0

2. Solve for k by factoring.

k² + 11k + 28 = 0

(k + 4)(k+7) = 0

k + 4 = 0
k = -4

k + 7 = 0
k = -7

Solution set: -4, -7

Is the solution set true?

1. Test k = -4 by plugging it in to the original equation.

k² + 11k = -28
(-4)² + 11(-4) = -28
16 + -44 = -28
-28 = -28

k = -4 is true.

2. Test k= -7 by plugging it in to the original equation.

k² + 11k = -28
(-7)2 + 11(-7) = -28
49 + -77 = -28
-28 = -28

3. x-intercepts

Definition: An x-intercept is where a function and the x-axis intersect.

Usually, x-intercepts are used in the context of graphing, as in "Look at the graph and find the x-intercepts."

4. Zeros/Roots

Definition: A zero, or root, describes the input of a function when the output is 0. A quadratic function will have 0, 1, or 2 zeros or roots.

Find the zero(s) or root(s) or the following function:

y = 3a² -10a + 8

1. Set the function equal to 0. Replace y with 0.

0 = 3a² – 10a + 8

2. Solve by factoring.

0 = (3a + -4)(a + -2)

Set the factors equal to 0 and solve for a.

3a + - 4 = 0
3a + -4 + 4 = 0 + 4
3a = 4
3a/3 = 4/3
a = 4/3

a + -2 + 2 = 0 + 2
a + -2= 0
a = 2

3. Verify that when a = 4/3, y = 0

0 = 3a² – 10a + 8
0 = 3(4/3)² -10(4/3) + 8
0 = -24/3 + 8
0 = 0

4. Verify that when a=2, y = 0

0= 3a² – 10a + 8
0= 3(2)² - 10(2) + 8
0= 12 – 20 + 8
0 = 0