Download Lesson 9.5

Lesson 9.5
Objective: To solve quadratic equations using
the quadratic formula.
What formula can be used to solve any quadratic equation?
Quadratic formula: When
Then the value
of x is…
ax  bx  c  0
2
 b  b  4ac
x
2a
2
Use the quadratic equation to solve for x.
a=1
2
x  9 x  14  0 b = 9
c = 14
Example:
 b  b 2  4ac
x
2a
 9  92  4 1 14
x
2 1
 9  81  56
x
2
 9  25
x
2
95
x
2
4
95

 2
2
2
 9  5  14

 7
2
2
Example:
Solve for x
x  5x  6  0
2
 b  b 2  4ac
x
2a
a=1
b=5
c = -6
 5  52  4 1  6
x
2 1
 5  25  24
x
2
 5  49
x
2
57
x
2
57
x
2
57
x
2
2
 1
2
 12

 6
2
Example:
Solve for x
 2x  6x  9  0
2
 6  62  4  2  9
x
2  2
 6  36  72
x
4
 6  108
x
4
 6  36 3
x
4
66 3
x
4
33 3
x
2
Simplify
Simplified
a = -2
b=6
c=9
Vertical Motion Model
A ball is thrown upwards with an initial velocity of 90 feet per
second from a height of 6 feet. Use the vertical motion model to
determine the time it will take the ball to hit the ground.
h = height of ground
t = time
v = initial velocity
s = starting height
a = -16 b = 90 c = 6
Applications of the Discriminant
The discriminant is the expression inside the radical in the
quadratic formula, b2
– 4ac.
• If b2 – 4ac is positive, then the equation has two solutions.
• If b2 – 4ac is zero, then the equation has one solution.
• If b2 – 4ac is negative, then the equation has no real solution.
The discriminant also tells the number of times the
parabola crosses the x-axis
Positive discriminant: The parabola crosses x-axis twice.
Zero discriminant: The parabola crosses x-axis once.
Negative discriminant: The parabola never crosses x-axis.
Positive
Two
solutions
Zero
One
Negative
No solutions
Examples: Find the discriminant and determine the number of solutions
1.
0  x 2  3x  4
a =1
b =-3
c =-4
b 2  4ac
 32  4  1  4
9 - -16
25 Two solutions
2
.
0  x  2x  5
2
a =1
b =2
c =5
b 2  4ac
22  4 1 5
4 – 20
–16 No solutions
3.
0  x2  4x  4
b 2  4ac
42  4 1  4
16 – 16
0
One solution