Transcript 1.8 Quadratic Formula & Discriminant

1.8 Quadratic Formula &
Discriminant p. 58
How do you solve a quadratic equation
if you cannot solve it by factoring,
square roots or completing the square?
Is there a fast way to decide?
Quadratic Formula
If you take a quadratic equation in standard
form (ax2+bx+c=0), and you complete the
square, you will get the quadratic formula!
 b  b  4ac
x
2a
2
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.)
Quadratic Formula : the one with the song!
It’s a way to remember the formula…
 b  b  4ac
x
2a
2
To use the quadratic formula...

Example – Two real solutions
3x2+8x=35
3x2+8x-35=0
a=3, b=8, c= -35
1.
 (8)  (8)  4(3)(35)
x
2(3)
 8  22 14 7
x


6
6 3
2
 8  64  420
x
6
 8  484
x
6
 8  22
x
6
OR
 8  22  30
x

 5
6
6
Two real solutions
What does an answer with two real solutions tell
you about the graph of the equation?
It tells you the two places that the graph
crosses the x axis (x-intercepts)
Example – One real solution
Solve 25x2 – 18x = 12x – 9.
25x2 – 18x = 12x – 9.
Write original equation.
Write in standard form.
25x2 – 30x + 9 = 0.
x = 30 + (–30)2– 4(25)(9) a = 25, b = –30, c = 9
2(25)
30 + 0
x = 50
x = 53
ANSWER
The solution is 3
5
Simplify.
Simplify.
One solution
CHECK
Graph y = –5x2 – 30x + 9 and
note that the only x-intercept is
0.6 = .
3

5
Example - Imaginary solutions
2.
-2x2=-2x+3
-2x2+2x-3=0
a=-2, b=2, c= -3
 (2)  (2) 2  4(2)( 3)
x
2(2)
 2  4  24
x
4
 2   20
x
4
 2   1* 4 * 5
x
4
 2  2i 5
x
4
 2 2i 5
x

4 4
1 i 5
x 
2 2
Imaginary Solutions
What does an answer with imaginary solutions
tell you about the graph of the equation?
The graph will not go through or touch the xaxis.
Discriminant:


2
b -4ac
The discriminant tells you how many
solutions and what type you will have.
If the discrim: Is positive – 2 real solutions
Is negative – 2 imaginary
solutions
Is zero – 1 real solution
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

9x2+6x-4=0
a=9, b=6, c=-4
b2-4ac=(6)2-4(9)(-4)
=36+144=180
2 real solutions
b.
c. 9x2+6x+5=0
a=9, b=6, c=5
b2-4ac=(6)2-4(9)(5)
=36-180=-144
2 imaginary solutions
A juggler tosses a ball into the air. The ball leaves the juggler’s hand
4 feet above the ground and has an initial vertical velocity of 40 feet
per second. The juggler catches the ball when it falls back to a
height of 3 feet. How long is the ball in the air?
Because the ball is thrown, use the model h = –16t2 + v0t + h0. To find
how long the ball is in the air, solve for t when h = 3.
h = –16t2 + v0t + h0
Write height model.
3 = –16t2 + 40t + 4
Substitute 3 for h, 40 for v0,
and 4 for h0.
0 = –16t2 + 40t + 1
Write in standard form.
t = – 40+
Quadratic formula
402– 4(– 16)(1)
2(– 16)
– 40+
1664
t=
– 32
t – 0.025 or t
2.5
Simplify.
Reject the solution – 0.025 because the ball’s time in the air cannot be
negative. So, the ball is in the air for about 2.5 seconds.
Assignment
Page 58, 3-48 every third problem,
52-54