Download The Quadratic Formula and the Discriminant

The Quadratic Formula and the Discriminant
Lesson 6-4
Here is the quadratic formula-which is proved by
completing the square.
The Quadratic Formula
__________________________________.
If
ax² + bx + c = 0,
Then
Example 1. Use the quadratic formula to solve this
quadratic equation:
3x² + 5x − 8 = 0
Solution. We have: a = 3, b = 5, c = −8.
Therefore, according to the formula:
x
=
−5 + 11
6
or
−5 − 11
6

6 16
or
6
6
These are the two __________ or __________________.
x  1or 
8
3
And they are__________ .
When the roots are rational,
we could have solved the equation by factoring, which is always the simplest method.
3x² + 5x − 8
x
=
=
(3x + 8)(x −1 )
−
8
3
or 1.
Example 2. Use the quadratic formula to find the roots of
each quadratic.
a) x² − 5x + 5
a=
b =
, c =
b) 2x² − 8x + 5
a= , b = , c =
c) 5x² − 2x + 2
a = , b = , c =
Discriminant
Copy the chart p. 356 onto formula sheet
The radicand b² − 4ac is called the discriminant.
• If the discriminant is
• a) Positive:
• and a perfect square 2 real, rational
• and is not a perfect square 2 real, irrational
• b) Negative: The roots are 2 imaginary
• c) Zero: 1 real
Example 3: Find the discriminant and find the nature of the
roots.
a.
4 x 2  25  20 x
a= ,b= ,c=
Describe the nature of the roots.
b.
3x 2  2  5 x
a= , b = , c =
Describe the nature of the roots