The Quadratic Formula
The Quadratic Formula is a general method to solve ANY quadratic equation. The trick to using the
quadratic formula is to substitute in for a, b, and c correctly and to follow your order of operations
(PEMDAS)!
Remember the general formula for a quadratic equation is ------ y  ax 2  bx  c
The Quadratic Formula: x 
Example #1
x 2  4x  3  0
b  b 2  4ac
,a0
2a
x 2  4x  3
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
o zero.
a  1,b  4,c  3
Identify you’re a, b, and c values.
(4)  (4)2  4(1)(3)
x
2(1)
Substitute the values of a, b, and c into the
Quadratic Formula.
x
(4)  16  12
2(1)
(4)  4
2(1)
42
x
2
x
42
2
6
x
2
x3
42
x
2
2
x
2
x 1
x
Use your order of operations to simplify
First work inside the square root sign multiply
out 4ac and simplify (-4)2.
Then add the two terms in your square root sign.
If you can take the square root do so, if not leave
the number in the square root sign.
Simplify the top of your fraction
Simplify the bottom of your fraction
Simplify and Solve for your first solution
Simplify and Solve for your second solution
Example #2
2 x 2  3x  5  0
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
to zero.
Identify you’re a, b, and c values.
Plug in your values for a, b, & c into your
Quadratic Formula.
Simplify using your order of operations.
Start inside the square root sign first.
Simplify and Solve for your first solution
Simplify and Solve for your second solution
Your Turn #1
3x 2  6x  9  0
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
to zero.
Identify you’re a, b, and c values.
Plug in your values for a, b, & c into your
Quadratic Formula.
Simplify using your order of operations.
Start inside the square root sign first.
Simplify and Solve for your first solution
Simplify and Solve for your second solution
Your Turn #2
x 2  4x  2  0
**** Save for Last!! ****
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
to zero.
Identify you’re a, b, and c values.
Plug in your values for a, b, & c into your
Quadratic Formula.
Simplify using your order of operations.
Start inside the square root sign first.
Simplify and Solve for your first solution
Simplify and Solve for your second solution
Your Turn!
3v2  5v  8
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
to zero.
Identify you’re a, b, and c values.
Plug in your values for a, b, & c into your
Quadratic Formula.
Simplify using your order of operations.
Start inside the square root sign first.
Simplify and Solve for your first solution
Simplify and Solve for your solve solution
The Quadratic Formula - Answer Key
Example #2
2 x 2  3x  5  0
2
2𝑥 + 3𝑥 − 5 = 0
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
to zero.
Identify you’re a, b, and c values.
𝑎 = 2, 𝑏 = 3, 𝑐 = −5
𝑥=
−(3) ±
√(3)2
− 4(2)(−5)
2(2)
𝑥=
−3 ± √9 + 40
4
𝑥=
−3 ± 7
4
Plug in your values for a, b, & c into your
Quadratic Formula.
Simplify using your order of operations.
Start inside the square root sign first.
Simplify and Solve for your first solution
𝑥=
−3 + 7
4
𝑥=
4
4
𝑥=1
Simplify and Solve for your second solution
−3 − 7
𝑥=
4
𝑥=
−10
4
𝑥=
−5
2
3x 2  6x  9  0
Your Turn #1
2
−3𝑥 + 6𝑥 + 9 = 0
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
to zero.
Identify you’re a, b, and c values.
𝑎 = −3, 𝑏 = 6, 𝑐 = 9
𝑥=
−(6) ±
√(6)2
− 4(−3)(9)
2(−3)
𝑥=
−6 ± √36 + 108
−6
𝑥=
−6 ± √144
−6
𝑥=
−6 ± 12
−6
Plug in your values for a, b, & c into your
Quadratic Formula.
Simplify using your order of operations.
Start inside the square root sign first.
Simplify and Solve for your first solution
−6 + 12
𝑥=
−6
𝑥=
6
−6
𝑥 = −1
Simplify and Solve for your second solution
−6 − 12
𝑥=
−6
𝑥=
−18
−6
𝑥= 3
x 2  4x  2  0
Your Turn #2
𝑥 2 + 4𝑥 + 2 = 0
**** SAVE FOR LAST!!!! ****
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
to zero.
Identify you’re a, b, and c values.
𝑎 = 1, 𝑏 = 4, 𝑐 = 2
𝑥=
−(4) ±
√(4)2
2(1)
𝑥=
−4 ± √16 − 8
2
𝑥=
−4 ± √8
2
𝑥=
−4 ± 2√2
2
𝑥=
2(−2 ± √2)
2
− 4(1)(2)
Plug in your values for a, b, & c into your
Quadratic Formula.
Simplify using your order of operations.
Start inside the square root sign first.
Simplify and Solve for your first solution
𝑥=
2(−2 + √2)
2
𝑥 = −2 + √2
Simplify and Solve for your second solution
𝑥=
2(−2 − √2)
2
𝑥 = −2 − √2
3v2  5v  8
Your Turn!
2
3𝑣 + 5𝑣 − 8 = 0
Make sure that all the terms of the quadratic
equation are on one side of the equation and set
to zero.
Identify you’re a, b, and c values.
𝑎 = 3, 𝑏 = 5, 𝑐 = −8
𝑣=
−(5) ±
√(5)2
− 4(3)(−8)
2(3)
𝑣=
−5 ± √25 + 96
6
𝑣=
−5 ± √121
6
𝑣=
−5 ± 11
6
Plug in your values for a, b, & c into your
Quadratic Formula.
Simplify using your order of operations.
Start inside the square root sign first.
Simplify and Solve for your first solution
−5 + 11
𝑣=
6
𝑣=
6
6
𝑣=1
Simplify and Solve for your second solution
−5 − 11
𝑣=
6
𝑣=
−16
6
𝑣=
−8
3