1. 5.6 Quadratic Formula & Discriminant

2. Quadratic Formula (Yes, it’s the one with the song!)
If you take a quadratic equation in standard form (ax2+bx+c=0), and you complete the square, you will get the quadratic formula!

3. When to use the Quadratic Formula
Use the quadratic formula when you can’t factor to solve a quadratic equation
(or when you’re stuck on how to factor the equation).

4. Examples
1. 3x2 + 8x = 35
3x2 + 8x – 35 = 0
a = 3, b = 8, c = -35 OR

5. 2. -2x2 = -2x + 3
-2x2 + 2x – 3 =0
a = -2, b = 2, c = -3

6. 3. 4x2 = 20x - 25
4x2 - 20x + 25 = 0
a = 4, b = -20, c = 25

7. Discriminant: b2-4ac
The discriminant tells you how many solutions and what type you will have.
If the discriminant:
Is positive – 2 real solutions
Is negative – 2 imaginary solutions
Is zero – 1 real solution

9. Examples 9x2+6x-4=0 a=9, b=6, c=-4 b2-4ac=(6)2-4(9)(-4) =36+144=180
2 real solutions
c. 9x2+6x+5=0
a=9, b=6, c=5
b2-4ac=(6)2-4(9)(5)
=36-180=-144
2 imaginary solutions
Find the discriminant and give the number and type of solutions.
9x2+6x+1=0
a=9, b=6, c=1
b2-4ac=(6)2-4(9)(1)
=36-36=0
1 real solution

10. Deriving the Quadratic Formula by Completing the Square
ax2 + bx + c = 0
Standard Form
ax2 + bx = -c
Isolate x’s on one side
Divide each term by a
Complete the square
Write as a binomial square
Find common denominator

11. The Quadratic Formula!!
Combine fractions on the right
Take the square root of both sides
Isolate the x
Separate the radical
Simplify the denominator
Combine the fractions