Download Solving Quadratic Functions

Exploring Quadratic Functions
and Inequalities
Advanced Algebra
Chapter 6
Solving Quadratic Functions
 Solve the following equation.
x  42  25
Solution:
x  42
 25
x  4  5
Square of
a Binomial
x  4  5
x45
x  9
x 1
Solving Quadratic Functions
 Multiply the following expressions.
x  3x  3  x 2  6 x  9
2x  52x  5  4 x 2  20 x  25
 Is there a pattern?
2×product of
both terms
 Shortcut Method
( x + 6 )2
1st term
last term
=
x2 + 12x + 36
square of
1st term
square of
last term
Solving Quadratic Functions
 Try using the shortcut method with these.
x  12 
2
x2  2x 1
4
4
2

2
x

x

x




3
9
3

 Now Try Backwards:
x2 + 8x + 16 = ( x  4 )2
x2 – 4x + 4 = ( x  2 )2
x2 + x + ¼ = ( x + ½ )2
Solving Quadratic Functions by
Completing the Square
 For example, solve the following equation by
completing the square. x 2  3x  18  0
Step 1  Move the constant to the other side.
x 2  3x  ___  18
Step 2  Notice the coefficient of the linear term is 3,
or b = 3. Therefore,
b
 
2
2
is the new constant needed to
create a Square Binomial. Add this value to both sides.
2
3
3
2
x  3 x     18   
2
2
2
Solving Quadratic Functions by
Completing the Square
Step 3  Factor and Solve.
2
3
3
x 2  3 x     18   
2
2
2
x
3 9

2 2
x
2
x
9 3

2 2
9 3
x 
2 2
2
6
x 3
2
3
72 9

x





2
4 4

3
81

x




2
4

3
9

x





2
2


3
9

2
2
12
x    6
2
Quadratic Formula
 Another way to solve quadratic equations is to
use the quadratic formula.
 b  b  4ac
x
2a
2
 This is derived from the standard form of the
equation ax2 + bx + c = 0 by the process of
completing the square.
Quadratic Formula
 The Quadratic Formula
 b  b  4ac
x
2a
2
The Discriminant
 The value of the discriminant, b2 – 4ac,
determines the nature of the roots of a quadratic
equation.
Discriminant  b2 – 4ac
b2 – 4ac
Value
Description
b2 – 4ac = 0
b2 – 4ac < 0
b2 – 4ac > 0
is a perfect
square
Intersects
the x-axis
once.
One real
root.
Does not
intersect
the x-axis.
Two
imaginary
roots.
Intersects
the x-axis
twice.
Two real,
irrational
roots.
Intersects
the x-axis
twice.
Two real,
rational
roots.
10
y
10
y
5
Sample
Graph
-5
y
5
5
x
x
-5
x
-5
-1
5
x
-5
-1
5
-6
y
5
-5
Solving Quadratic Functions with
the Quadratic Formula
 For example, solve the following equation with
the quadratic formula. 4 x 2  25  20 x
Step 1  Write quadratic equation in Standard Form.
The discriminant, (–20)2 – 4(4)(25) = 0.
There is one real, rational root.
4 x  20 x  25  0
2
Step 2  Substitute coefficients into quadratic formula.
In this case a = 4, b = –20 and c = 25
x
  20 
 20
24
2
 4425
20 5
x

8 2
Solving Quadratic Functions with
the Quadratic Formula
 For example, solve the following equation with
the quadratic formula.
3x 2  2  5 x
Step 1  Write quadratic equation in Standard Form.
3x  5 x  2  0
2
The discriminant, (–5)2 – 4(3)(2) = 1.
There are two real, rational roots.
Step 2  Substitute coefficients into quadratic formula.
In this case a = 3, b = –5 and c = 2
x
  5 
 5
23
2
 432
6
x  1
6
4 2
x 
6 3
Homework