This document contains information about solving quadratic equations using the quadratic formula. It provides the formula, an explanation of how it is derived using completing the square, and several examples of solving quadratic equations using the formula. It also discusses the discriminant and how it relates to the number of real roots a quadratic equation will have.

1.  John 6:47
Truly, truly, I say to you, whoever believes has
eternal life.
 Isaiah 33:22
For the LORD is our judge, the LORD is our
lawgiver, the LORD is our king; he will save
us.

2. The Quadratic Formula

3. Quadratic Formula
IF
0 2 ax bx  c 
THEN
4 2   

b b ac
a
x
2

4. Proving the Quadratic Formula
 Solve by completing the square
ax2 bx  c  0
c
a
b
x    2
x
a
2 2
2
b
b
a

 
2 4a




c
a
b
2
x     2
a
b
a
b
x
a
2
2
2
4 4
c
a
2 2
b

x   
a
b
a




 2
2 4
b ac
2
2 2
4
4
b
a

2 a
x

 





b ac
a
b
a
x
2
4
2
2 
  
2 2
4
x   
2 2
b
b
2 4 4
a
ac
a
a




2 LCM  4a
b ac
b
4 2   
b b ac
a
x
2

a
a
x
2
4
2
2 
  

5. #1 Solve using the quadratic formula.
3 7 2 0 2 x  x  
a  3, b  7, c  2
4 2   

b b ac
a
x
2
( 7) ( 7) 4(3)(2) 2     
2(3)
x 
7  49  24
6
x 
7  25
7  5
6
x 
12
6
x 
2
6
x 
1
3
x  2,
6
x 

6. #2 Solve using the quadratic formula
2 4 5 2 x  x 
a  2, b  4, c  5
4 2   

b b ac
a
x
2
( 4) ( 4) 4(2)( 5) 2      
2(2)
x 
4  16  40
4
x 
4  56
4
x 
4  414
4
4  2 14
4
x 
x 
2 4 5 0 2 x  x  
14
2
x 1
x  2.9 x  0.9

7. Solve using the quadratic formula
3.
4.
5.
5 7 2 x  x  
3m 8 10m 2  
3
5
x i
2
  
2
2
3
m  4, 
x 64 16x 2  
x 8

8. #3 Solve using the quadratic formula
5 7 2 x  x  
a 1, b  5, c  7
4 2   

b b ac
a
x
2
5 5 4(1)(7) 2   
2(1)
x 
 5  25  28
2
x 
 5   3
2
x 
3
5
x i
2
  
2
5 7 0 2 x  x  
3
5
x i
2
  
2

9. #4 Solve using the quadratic formula
3m 8 10m 2  
a  3, b  10, c  8
4 2   

b b ac
a
m
2
( 10) ( 10) 4(3)( 8) 2      
2(3)
m 
10  100  96
6
m 
10  196
10 14
6
m 
24
6
m 
4
6
m  
2
3
m  4, 
3 10 8 0 2 m  m 
6
m 

10. #5 Solve using the quadratic formula
x 64 16x 2  
a 1, b  16, c  64
4 2   

b b ac
a
x
2
16 16 4(1)(64) 2   
 
2(1)
x 
16  256  256
2
x 
16  0
2
x 
x 8
16 64 0 2 x  x  

11. The Discriminant
 Discriminant  In the quadratic
formula, the expression
underneath the radical
that describes the
nature of the roots.

12. Understanding the discriminant
Discriminant
ac b 4 2 
# of real roots
4 0 2 b  ac  2 real roots
4 0 2 b  ac  1 real roots
4 0 2 b  ac  No real roots

13. #6 Using the discriminant
4 3 1 0 2 y  y  
a  4, b  3, c  1
discriminant b 4ac 2  
discrimina nt ( 3) 4(4)( 1) 2    
discriminant  916
discriminant  25
25  0
2 real roots

14. #7 Using the discriminant
4 5 2 x  x 
a  4, b  1, c  5
discriminant b 4ac 2  
discrimina nt ( 1) 4(4)(5) 2   
discriminant  180
discriminant  79
79 0
no real roots

15. Using the discriminant
8.
9.
2 4 5 2 x  x 
4 8 4 2 x  x 
discriminant  56
2 real roots
discriminant  0
1 real root

16. #8 Using the discriminant
2 4 5 2 x  x 
2 4 5 0 2 x  x  
discriminant b 4ac 2  
discrimina nt ( 4) 4(2)( 5) 2    
discriminant  1640
discriminant  56
56  0
2 real roots

17. #9 Using the discriminant
4 8 4 2 x  x 
4 8 4 0 2 x  x  
discriminant b 4ac 2  
discrimina nt ( 8) 4(4)(4) 2   
discriminant  6464
discriminant  0
0  0
1 real root