Quadratic Formula

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