Download Solving Quadratic Equations by Completing the Square

Quadratic Formula
Objective: TSW solve quadratic equations using the quadratic formula
http://www.youtube.com/watch?v=z6hCu0EPs-o
http://www.youtube.com/watch?v=YCuXiujC3KE
THE QUADRATIC FORMULA
1. When you solve using completing the square
on the general formula ax 2  bx  c  0
you get:
b  b 2  4ac
x
2a
2. This is the quadratic formula!
3. Just identify a, b, and c then substitute into
the formula.
WHY USE THE
QUADRATIC FORMULA?
The quadratic formula allows you to solve
ANY quadratic equation, even if you
cannot factor it.
An important piece of the quadratic formula
is what’s under the radical:
b2 – 4ac
This piece is called the discriminant.
WHY IS THE DISCRIMINANT
IMPORTANT?
The discriminant tells you the number and types of answers
(roots) you will get. The discriminant can be +, –, or 0
which actually tells you a lot! Since the discriminant is
under a radical, think about what it means if you have a
positive or negative number or 0 under the radical.
WHAT THE DISCRIMINANT
TELLS YOU!
Value of the Discriminant
Nature of the Solutions
Negative
2 imaginary solutions
Zero
1 Real Solution
Positive – perfect square
2 Reals- Rational
Positive – non-perfect
square
2 Reals- Irrational
Example #1
Find the value of the discriminant and describe the nature of the
roots (real,imaginary, rational, irrational) of each quadratic
equation. Then solve the equation using the quadratic formula)
1.
2 x 2  7 x  11  0
Discriminant =
a=2, b=7, c=-11
b 2  4ac
(7) 2  4(2)(11)
49  88
Discriminant =
137
Value of discriminant=137
Positive-NON perfect
square
Nature of the Roots –
2 Reals - Irrational
Example #1- continued
Solve using the Quadratic Formula
2 x 2  7 x  11  0
a  2, b  7, c  11
b  b 2  4ac
2a
7  7 2  4(2)(11)
2(2)
7  137
4
2 Reals - Irrational
Complex Solutions
Coefficients are:
Note that
The Discriminant is
negative:
Use the Quadratic
Formula:
a = 5, b = 2, c = 1
b2 - 4ac = 22 - 4×5×1 = -16
x = [ -2 ± √(-16) ] / 10
The square root of -16 is 4i
(i is √-1, read Imaginary Numbers to find out more)
So:
x = ( -2 ± 4i )/10
Answer: x = -0.2 ± 0.4i
The graph does not cross
the x-axis. That is why we
ended up with complex
numbers.
Solving Quadratic Equations
by the Quadratic Formula
Try the following examples. Do your work on your paper and then check
your answers.
1. x  2 x  63  0
2
2. x  8 x  84  0
2
3. x  5 x  24  0
2
4. x  7 x  13  0
2
5. 3 x 2 5 x  6  0
1.  9, 7 
2.(6, 14)
3.  3,8 
 7  i 3 
4. 

2


 5  i 47 
5. 

6

