Learn How to Complete the Square by using equations

Technical Education

Learn How to Complete the Square by using completing the square method is a way of solving quadratic equation.

It is so simple if you understand how we derived our formula method.

Remember quadratic Equations are polynomial of the second degree order and is form can be represented as below:

Ax^2 + Bx+ C =0

Some quadratics is very simple to solve because they come in simple form like below:

Say (x-3)^2=9

This type of quadratic equation could quickly be solved by taking square root of both sides of the equation.

i.e sqrt ( x-3)^2 =sqrt(9)

x-3=+0r-3 (note that when you take a square root of a number say 9 for example,the result would be either + 0r – )

By solving for x in the equation above we are going to have two answers.

i.e x=3+3 or x=3-3

x=6 or x=0

What about the situation when our equations do not come in this form? Most quadratic equations will not come neatly squared like this. In this case you first use your mathematical technique to arrange the quadratics in a neatly squared part equals to a number like the example treated above. Completing the square method as per given below:

For a typical Example:

Solve the quadratic equation 4x^2 -2x-5=0


Step 1: move -5 to the R.H.S to the equation (R.H.S-right hand side)

4x^2-2x=5 (remember when you move -5 to the other side of the equation it becomes +5)

Step 2: Divide through by the co efficient of your X squared term (which is 4 in our example)

The equation now becomes:

X^2 – ½X = 5/4

Step 3: Take half the coefficient of X term and square it and add it to both sides

½ of -1/2 =-1/4

When you square it you have 1/16 add to both side of the equation which now becomes:

X^2 – 1/2X + 1/16 = 5/4 + 1/16

Step 4: Convert the left hand side to a squared form and simplify the R.H.S

(x-1/2)^2 = 21/16 (now you have a simple squared form just like our first example)

Step 5: Find the square root of both sides

x-1/2 = + or – sqrt(21/16)

solving for x finally leads to 2 answers either:

X=1/2- sqrt(21/16) or X= ½ + sqrt(21/16)

Congratulations you have successfully complete the steps for solving a quadratic equation using completing the square method.


  1. Move the number part to the right hand side of the equation


  1. Divide through by the coefficient of the x squared term
  2. Take one-half the coefficient of the x term, square it and add it to both sides of the equation
  3. Re-arrange your equation by putting the right hand side in squared form and simplifying the left hand side. Take the square root of both sides remembering the + or – sign on the right hand side. Finally solve for two possible values of X


Solve X^2 +6X-7=0 by completing the square method.

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