The Quadratic Formula & Discriminant

Essential question – How do you solve a quadratic equation using the Quadratic Formula?

When to use the quadratic formula:

Where does the formula come from?

Discriminant: b2-4ac The discriminant tells you how many

solutions and what type you will have. If the discriminant b2-4ac

Is positive, there are 2 real solutions

(2 x-intercepts) Is negative, there are 2 imaginary solutions

(0 x-intercepts) Is zero, there is 1 real solution (double root!)

(1 x-intercept)

Examples Find the discriminant

and give the number and type of solutions.

a. 9x2+6x+1=0

a=9, b=6, c=1

b2-4ac=(6)2-4(9)(1)

=36-36=0

1 real solution

b. 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

Quadratic Formula(it has a song!)

Notice where the discriminant is in the quadratic formula!

a

acbbx

2

42

Examples1. 3x2+8x=35

3x2+8x-35=0

a=3, b=8, c= -35

6

420648 x

)3(2

)35)(3(4)8()8( 2 x

6

4848x

6

228x

3

7

6

14

6

228

x

OR

56

30

6

228

x

2. -2x2=-2x+3

-2x2+2x-3=0

a=-2, b=2, c= -3

)2(2

)3)(2(4)2()2( 2

x

4

2442

x

4

202

x

4

5*4*12

x

4

522

i

x

4

52

4

2

i

x

2

5

2

1 ix

3. 4x2+10x=-10x-25

4x2+20x+25=0

a=4, b=20, c=25

2(20) (20) 4(4)(25)

2(4)x

20 400 400

8x

20 0

8x

5

2x

Now you try…

p. 117, #2-16 even