Completing the Square
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Transcript Completing the Square
13.3 Completing the Square
• Objective: To complete a square for a quadratic equation
and solve by completing the square
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
To Complete the Square
x2 + 6x
• Take half of the coefficient of ‘x’ 3
• Square it and add it 9
x2 + 6x + 9 = (x + 3)2
Complete the square, and show what the perfect
square is:
x 12x x 12x 36
2
2
y 14 y
y 14 y 49
y 10 y
y 10y 25
2
2
x 5x
2
2
2
25
x 5x
4
2
x 6
2
y 7
2
y 5
2
5
x
2
2
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 (today’s lesson)
• 2.) Quadratic Formula (Next week’s lesson)
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
Solve by Completing the Square
x 6 x 16 0
2
x 6 x 16
2
+9
+9
x 6 x 9 25
2
x 3 25
x 3 5
x 3 5 x 8 x 2
2
Solve by Completing the Square
x 22 x 21 0
2
x 22 x 21
2
+121
+121
x 22 x 121 100
2
x 11 100
x 11 10
x 11 10 x 21 x 1
2
Solve by Completing the Square
x 2x 5 0
2
x 2x 5
2
+1
+1
x 2x 1 6
2
x 1 6
x 1 6
2
x 1 6
Solve by Completing the Square
x 10 x 4 0
2
x 10 x 4
2
+25
+25
x 10 x 25 29
2
x 5 29
2
x 5 29
x 5 29
Solve by Completing the Square
x 8 x 11 0
2
x 8 x 11
2
+16
+16
x 8 x 16 5
2
x 4 5
x 4 5
2
x 4 5
Solve by Completing the Square
x 6x 4 0
2
x 6 x 4
2
+9
+9
x 6x 9 5
2
x 3 5
x 3 5
2
x 3 5
The coefficient of
2 x 3x 3 0
2
2
2
2
2
3
3
x x 0
2
2
2
x
must be “1”
2
3 33
x
4 16
2
3
33
x
4
16
3
3 3 2 3
2
x x
4
33
3
33
2
2 2
x 3 33
x
4
4
16
3
9
3
9
2
x
2
x
16
2 16
4
The coefficient of
3x 12 x 1 0
1
2
x 4x
3
2
x
must be “1”
2
1
x 4 x 4 4
3
2
x 2
2
11
3
11
x2
3
11
x 2
3
33
x 2
3
3
3
6
x 32
33
3
6 33
x
3