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