Completing the Square – Formula, Method and Examples

Completing the Square is a procedure in which any quadratic equations are transformed into the vertex form. In simple words, Completing the square is the technique of converting axe*x + bx + c into the ( x- h ) * ( x- h) + k. 

Formula of Completing the Square

The Formula for Completing the Square is,

axe * x + b * x + c = a ( x + m ) * ( x + m )+ n

Here,

n= constant value

m = a real value

Applications of Completing the Square

The most popular application of using the procedure of completing the square is solving any problems of quadratic equations. One can also develop the formula for quadratic problems. With the assistance of this technique of completing the square, the graphing of the equations of quadratics becomes simple and manageable. We can also get the peak value and the minimum value of any quadratic equation. 

How do Get the Value of Any Quadratic Equation Using the Method of Completing the Square?

To solve any problems using the technique of completing the square, one has to follow certain steps. Those steps are as follows;

Suppose that there is an equation of 

axe*x + bx + c 

• First of all, we have to get the value of “m”. 
• To get ” m”, there is a formula that is m = b/2a. 
• Then, we have to get the value of “n”. To get ” n”, there is a formula that is,
  n = c – b*b / 4a.
• Then in the next step, we have to apply the values in the formula illustrated,

axe*x + bx + c = an ( x+m ) * ( x+m ) +n

• The value got, will be the solution to the problem given.

Examples of Solving the Quadratic Equations

Now, we will be illustrating and discussing a few examples to solve the quadratic equations through the assistance of completing the square. Those examples are:

Example 1: There is a Quadratic Equation,
x*x – 7x. Get the solution to this particular equation, using the procedure of Completing the Square.

Solution:

The quadratic equation is given,

x*x – 7x.

This particular equation is in the form of axe*x + bx + c,

a = 3, b= -7 and c = 1,

Now we have to get the value of m and n,

To get m, the formula is m=b/2a

And to get n, the formula is n=c-b*b/4a

m = -⅓ and n = ⅔

Now,

By applying all these values to the prime formula, we will get,

axe*x +b*x+c = 3(x+ -⅓* -⅓ ) + ⅔

After solving this particular equation, we will have the value.

Example 2: There is another Quadratic Equation,
x*x- 2*x-8 =0. We have to get the solution to this specific equation by taking the assistance of the procedure of Completing the Square.

Solution:

The Quadratic Equation given is,

x*x -2x -8,

This given equation is in the form of axe*x+bx + c,

Here, a = 1, b= -2, and c= -8,

Now, we have to get the solution of m and n by applying the values to the formulas.

Then apply all the combined values to the main formula to get the solution of the equation.

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