Download 3.1 Solving Equations Using Addition and Subtraction

11/3/2014
Solving Quadratic
Equations (4 - 2)
MA
Definition of a Quadratic
Equation
Standard form of a quadratic equation is:
ax2 + bx + c = 0
Where a, b, and c are real numbers with a ≠ 0.
A quadratic equation in x is also called a
second-degree polynomial equation in x.
The Zero-Product Principle
If the product of two algebraic
expressions is zero, then at least one of
the factors is equal to zero.
If AB = 0, then A = 0 or B = 0.
Example 1
x2  7 x = 0
xx  7  = 0
x = 0 or x  7 = 0
x=7
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Example 2
Solve
2x2 + 7x = 4
2x2 + 7x - 4 = 0
(2x - 1)(x + 4) = 0
2x-1=0
2x=1
x=½
Example 3
or
x+4=0
x = -4
Solve
(2x - 3)(2x + 1) = 5
4x2 - 4x - 3 = 5
4x2 - 4x - 8 = 0
4(x2 – x - 2)=0
4(x – 2) (x + 1) = 0
x-2=0
x+1=0
x=2
x = -1
The Square Root Method
If u is an algebraic expression and d is a
positive real number, then u2 = d has
exactly two solutions.
If u2 = d, then u = ¯d or u = -¯
d
OR
If u2 = d then u = ¯
d
Example 4
x2  16 = 0
x2 = 16
x 2 =  16
x = 4
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Solve by completing the square.
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Example 6
b
2 – 11x - 4 = 0
 
3x
2
x – 6x + 7 = 0
2 2
2
Example 5
 11 2
 36 
  
22 
x2 – 6x + 9 = -7 + 9 
  32
2
 11 
x2 – 6x + 9 = 2
 2

 3

2
x2 – 6x + ___ = -7
(x – 3)
=2
 11 1 


 3 22 
(x  3)2 =  2
2
 11 


 6 
x 3 =  2
x =3 2
x=
1
or 4
3
3x 11x 4

=
3
3
3
11x
4
2
x 
 ___ =
3
3
11x 121 4 121
x2 

= 
3 2 36 3 36
48 121
 11 

x  =
6  36 36

2
11  169

x  =
6
36

11
13
x =
6
6
11 13
x= 
6 6
Quadratic Equation
ax 2  bx  c = 0
Quadratic Formula
 b  b 2  4ac
x=
2a
Example 7
2 x 2  3x  8 = 0
x=
  3 
=
 3  4  2  8
2  2
2
3  9   8 8 
4
3  9  64
=
4
=
3  73
4
3