Download Page 1 1.4 Quadratic Equations In this section we will discuss the

1.4 Quadratic Equations Definition A quadratic equation is a second degree polynomial equation that can be written in the standard form 0;
0 In this section we will discuss the following four methods of solving quadratic equations: a. The factoring method b. Using the square root principle c. Completing the square d. Using the quadratic formula The zero product rule: 0, then 0
0 If ·
1. Solve the equation 3 2
5
0 Solution: 3 2
5
0 3 0
2
5 0 3
2
5 3
Solution Set: 3,
Guidelines for Solving Quadratic Equations using the Factoring Method a. Rewrite the equation in standard form by isolating zero on one side of the equation. b. Factor the polynomial c. Solve the equation using the zero product rule In examples 2‐4, solve the given quadratic equation using the factoring method. 2. x 2  4 x  12
x 2  4 x  12  0
3. 2 x 2  7 x  4
2 x2  7 x  4  0
4. 4 x 2  2 x  0
2 x  2 x  1  0
 x  6  x  2   0
 2 x  1 x  4   0
2 x  0 or 2 x  1  0
x  6  0 or x  2  0
2 x  1  0 or x  4  0
x  0 or x 
x  6 or x  2
6, 2
2 x  1 or x  4
x
1
or x  4
2
1

 , 4 
2

 1
0, 
 2
1
2
The Square Root Principle: , then If √ or √ , Equivalently √ In example 5‐7, solve the given quadratic equation using the square root principle. 5. x 2  18
7.  2 x  8   27
2
6. 3 x 2  1  47
x   18
3 x 2  48
2 x  8   27
x  3 2
x  16
2 x  8  i 27
x   16
2 x  8  3i 3
2
3 2
4
2 x  8  3i 3
x  4 
3i 3
2
Guidelines for solving a quadratic equation by Completing the Square a. Isolate the constant term on one side of the equation b. Divide both sides of the equation by the coefficient of c. Add the square of one‐half the coefficient of x to both sides of the equation d. Rewrite the left‐hand side of the equation as e. Solve the resulting equation using the square root principle In examples 8‐11, solve the given quadratic equation by Completing the Square. (see the last page for a detailed example) 8. x 2  10 x  2  0
x 2  10 x  2
x 2  10 x  52  2  52
9. x 2  12 x  6  0
10. 2 x 2  5 x  9  0
x 2  12 x  6
x 2  12 x   6   6   6 
2
2x2  5x  9
x2 
2
2
x 2  10 x  25  27
 x  5
2
 27
x 2  12 x  36  6  36
 x  6
2
 30
15 5
 
22 4
5
9
x
2
2
2
5
5 9 5
x    
2
2 4
4
5
25 9 25 9 8 25 72 25
x2  x 
    


2
16 2 16 2 8 16 16 16
x2 
2
x  5   27
x  5  3 3
x  6   30
x  6  30
5  97

x  
4  16

x
5
97
97


4
16
4
5
97
x 
4
4
11. 3 x 2  2 x  4  0
3x2  2 x  4
x2 
1 2
2
1
     
2 3
6
3
2
4
x
3
3
2
2
2
4  1
 1
x  x     
3
3  3
 3
2
1 4 1 4 3 1 12 1 13
x2  x       
 
3
9 3 9 3 3 9 9 9 9
2
2
1
13

x  
3
9

x
1
13
13


3
9
3
x
1
13 1  13


3
3
3
Quadratic Formula 0 are given by: The solutions of a quadratic equation written in standard form b  b 2  4ac
x
2
a
In examples 12 & 13, solve the given quadratic equation using the quadratic formula. 12. 3 x 2  2 x  4  0
a  3, b   2, c  4
x






b 
b 2  4 ac
2a
 2  
2
4  48
6
2
 44
2  2 i 11
6
6
1  i 11
3
8 x 2  3 x  36
8 x 2  3 x  36  0
x
6

1 
2
12  x 2  x   12  3 
4 
3
  2   4  3  4 
2 3 
2
2 1  i 11
2 2 1
x  x3
3
4
First C lear fractions by m ultiplying bith sides by 12 (the LC D )
13.
x

x
x
b 
3 
b 2  4 ac
2a
3 2  4  8   36 
2 8 
3 
9  1152
2 8 
 3  1161  3  9(129)
 3  3 129


16
16
16
A detailed example on how to solve a quadratic equation by completing the square.