Completing the square

General steps for completing the square

The method of 'Completing the square' can be applied to Quadratic Equations that
are written in the standard form. This method uses the algebraic property of expansion
of the square of the sum of two terms, which means:


In this method, simple mathematical operations are applied to a Quadratic Equation
(written in the standard form) in order to make its LHS (Left Hand Side) expression
of the form a² + 2ab + b². Then the LHS expression is factored
to the form (a + b)² by applying the above mentioned property. The following
example will further clarify this method:
Equation:

2x² + 5x + 3

Solution:

Step 1 : Divide the equation throughout
by the coefficient of x-squared.

The coefficient of x-squared is 2, so dividing the equation throughout by 2,

Step 2 : Move the number/constant
term to the RHS (Right Hand Side).

The constant term is 3/2, so moving it to RHS,

Step 3 : Take the coefficient of
'x', divide it by 2, then square it. Add the resulting term to both sides of the
equation (That is, add it to LHS as well as RHS).

Coefficient of 'x' is 5/2. Dividing it by 2, we obtain 5/4. Squaring it, we obtain
25/16. Adding 25/16 to both sides,

Simplifying the RHS,

Step 4 : Compare the LHS with a²
+ 2ab + b² and factor it to (a + b)².

The LHS is now in the form of a² + 2ab + b², where 'a' is
'x' and 'b' is 5/2. Applying the property a² + 2ab + b² =
(a + b)², the LHS is written in the form (a + b)²,

Step 5 : Solve for 'x'.

Taking square root of both LHS and RHS,



Move '5/2' to RHS,

Either

or

In the above example, each step is done as it should be done with all other quadratic equations. In short, there are four steps required to be done in the method of completing the square:

• Divide the equation throughout by coefficient of x-squared
• Move the constant term to RHS
• Take the coefficient of 'x', divide it by 2, square it, and then add it to LHS and RHS of the equation
• Compare the LHS with a² + 2ab + b² and write it in the form of (a + b)²
• Solve for 'x' by taking square root on both sides.

There are additional solved examples to help clarify the above procedure here.