1. 4.6 The Quadratic Formula and the Discriminant
Objectives:
Solve quadratic equations by using the Quadratic Formula
Use the discriminant to determine the number and type of roots for a quadratic equation.

2. Quadratic Formula
Always works to solve a quadratic equation, but is a little lengthy. The solutions of a quadratic equation of the form where a≠0 are given by the formula:

3. Example
x²-8x=33 x²-8x-33=0 Set = 0 a=1, b=-8, c=-33

4. Another Example
7x²+6x+2=0 a=7, b=6, c=2 Since ALL of the coefficients are divisible by 2, simplify by dividing them by 2.

5. Discriminant
The discriminant describes the solution to a quadratic equation. The part of the quadratic formula under the radical is the discriminant or b²-4ac.
If b2 – 4ac > 0, and a perfect square
You have two rational roots
If b2 – 4ac >0, and not a perfect square.
You have two irrational roots
If b2 – 4ac = 0
You have 1 real, rational root. (Repeated root)
If b2 – 4ac < 0
You have two complex roots

6. Examples b. 7x²-3x=0 (-3)²-4(7)(0)= 9-0=9
Two rational roots because 9 is positive and a perfect square.
3x²-x+5=0
(-1)²-4(3)(5)=
1-60=-59
Two complex roots because the discriminant is a negative.
Examples
Find the discriminant and describe the number and type of roots.
x²-16x+64=0
b²-4ac
(-16)²-4(1)(64)= =0
One real, rational root because the discriminant equals 0.

7. We have discussed several methods for solving quadratic equations – which one do you use?
Can Be Used
When to Use
Graphing
sometimes
Use only if an exact number is not required. Best use to check the reasonableness of solutions found algebraically
Factoring
Use if the constant term is 0 or if the factors are easily determined
Square Root Property
Use for equations in which a perfect square is equal to a constant
Completing the Square
always
Useful for equations of the form ax2 + bx + c where b is even
Quadratic Formula
When other methods fail or are too tedious

8. Solve – use any method.
4. x²+5x+8=0 Doesn’t factor, not easily done by completing the square (5 is odd) so use quadratic formula.
1. 7x²-14x=0
7x(x-2)=0
x=0, x=2
x²-64=0
x²=64
x=8
x²-16x+64=0
(x-8)(x-8)=0
x=8

9. Homework Workbook Page 55