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COMPLETING THE SQUARE
Purpose: Another method to solve quadratic equations that will always work
To complete the square of: 𝐚𝐚𝐱𝐱 𝟐𝟐 + 𝐛𝐛𝐛𝐛 + 𝐜𝐜 = 𝟎𝟎
1. Isolate the terms containing x on the left side of the equation and the constant term on the
right.
2. If a ≠ 1, divide each term by a.
1
3. Take half of the middle coefficient (multiply by ).
2
4. Square the number found in step 3.
5. Add the number found in step 4 to both sides of the equation.
6. Factor the left side as a perfect square.
7. Simplify the right side.
NOTE: The equation should look like (x ± number)2 = NUMBER
8. Use the Square Root Method to solve the equation.
Steps to complete the square of:
1) Isolate the constant on one side of the Equation.
2) The coefficient of the x2 term must be one,
If it isn’t, divide both sides by that coefficient.
3) Take half of the middle coefficient and square it.
This is what is going to be added to both sides of
the equation.
4) Factor the left side and simplify the right side
NOTICE
5
6
=�
2
6
36
3x 2 + 5x = 6
5
x2 + x = 2
3
5 2
1 5
� � =→ � � =
2 3
6
5
25
3
�x + � =
36
6
5
x+ =±
6
√97
√36
25
36
25
x 2 + x + 36 = 2 + 36
5 2
25
5) Now apply the square root property and solve for x
��x + 5� = �97
3x 2 + 5x − 6 = 0
5 2
�x + � =
6
5
x=− ±
6
72
36
97
+
25
=
−5±√97
6
36
36
√97
6