Steps for completing the square and solving quadratics

1. Completing the Square
• Objective: To complete a square for a quadratic equation
and solve by completing the square

2. Steps to complete the square
• 1.) You will get an expression that looks like this:
AX²+ BX
• 2.) Our goal is to make a square such that we have
(a + b)² = a² +2ab + b²
• 3.) We take ½ of the X coefficient
(Divide the number in front of the X by 2)
• 4.) Then square that number

3. To Complete the Square
x2 + 6x
• Take half of the coefficient of ‘x’
• Square it and add it
3
9
x2 + 6x + 9 = (x + 3)2

4. Complete the square, and show what the perfect
square is:
xx 122
 36122
 xx  2
6x
yy 142
 49142
 yy  2
7y
yy 102
 25102
 yy  2
5y
xx 52

4
25
52
 xx
2
2
5






x

5. To solve by completing the square
• If a quadratic equation does not factor we can solve it by
two different methods
• 1.) Completing the Square
• 2.) Quadratic Formula

6. Steps to solve by completing the square
1.) If the quadratic does not factor, move the
constant to the other side of the equation
Ex: x²-4x -7 =0 x²-4x=7
2.) Work with the x²+ x side of the equation and
complete the square by taking ½ of the coefficient
of x and squaring
Ex. x² -4x 4/2= 2²=4
3.) Add the number you got to complete the square to
both sides of the equation
Ex: x² -4x +4 = 7 +4
4.)Simplify your trinomial square
Ex: (x-2)² =11
5.)Take the square root of both sides of the equation
Ex: x-2 =±√11
6.) Solve for x
Ex: x=2±√11

7. Solve by Completing the Square
2
6 16 0x x  
2
6 16x x 
+9 +9
2
6 9 25x x  
 
2
3 25x  
 3 5x   
3 5x   8x  2x  

8. Solve by Completing the Square
2
22 21 0x x  
2
22 21x x  
+121 +121
2
22 121 100x x  
 
2
11 100x  
 11 10x  
11 10x    21x   1x  

9. Solve by Completing the Square
2
2 5 0x x  
2
2 5x x 
+1 +1
2
2 1 6x x  
 
2
1 6x  
 1 6x   
1 6x  

10. Solve by Completing the Square
2
10 4 0x x  
2
10 4x x 
+25 +25
2
10 25 29x x  
 
2
5 29x  
5 29x   
5 29x   

11. Solve by Completing the Square
01182
 xx
1182
 xx
+16 +16
51682
 xx
  54
2
x
  54 x
54 x

12. Solve by Completing the Square
0462
 xx
462
 xx
+9 +9
5962
 xx
  53
2
x
  53 x
53x

13. The coefficient of x2 must be “1”
0332 2
 xx
0
2
3
2
32
 xx
16
33
4
3
2






x
2
3
2
32
 xx
4
3
2
2
3

2 9 9
16 16
3 3
2 2
x x  
16
33
4
3
x
16
33
4
3
x
2 2 2 2
33
4
4
333
x

14. The coefficient of x2 must be “1”
2
3 12 1 0x x  
2 1
4
3
x x  
2
4 4
1
4
3
x x   
 
2 11
2
3
x  
11
2
3
x   
11
2
3
x   3
3

33
2
3
x  

15. 33
2
3
x  
6
3
6 33
3
x

