## Contents:

This article covers one of the methods used to solve the quadratic functions which is called ” completing the square method”.

The following are the key concepts that this article covers in detail.

- What is a perfects square quadratic?
- General forms of a perfect square quadratic.
- How completing the square method is related to perfect square quadratic?
- What is the number to be added and subtracted to complete the square?
- Worked out examples solving a quadratic using “completing the square method”

# 1)Perfect square Quadratic.

A quadratic polynomial, which can be written as the product of two identical binomials, is called as a perfect square quadratic. This plays a key role in solving a quadratic equation using completing the square method.

### i)General forms of perfect square quadratic equations.

There are two general form of representing a quadratic equation as a perfect square .

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# 2)Solving the quadratic using “completing the square” method.

The phrase *“Completing the square”* conveys that the given quadratic equation has to be transformed into a *“perfect square quadratic”*.

The aim is to represent any arbitrary quadratic equation in the form of a perfect square quadratic.

In order to represent it, there is a need to introduce a constant number which helps us to write any quadratic as perfect square quadratic.

### ii)What is that number to be introduced to complete the square?

Let us consider the perfect square quadratic .We have the first terms as , middle term as and the last term as .

All we need is to re-write the given quadratic similar to this standard form ensuring that the leading coefficient of the Quadratic, ie the coefficient of , is 1. We compare this given quadratic with the standard form and get the coefficients a,b, c.

The constant number that is essential to complete the square is given by the formula .

We add and subtract this term to the given quadratic and group the first and middle term with the to complete the square.

**3)Worked out examples**

Let us start with a simple example.

**Strategy Step-by-Step:-**

**Step1**: First of all, replace y with 0.Identify the leading coefficient of the quadratic equation that is a.

- In the current problem, replacing y with 0, we have .On one to one comparison with the standard form the quadratic equation, we can say that the leading coefficient is a=1. .

**Step2: **Divide the equation throughout by the leading coefficient “a” to make the coefficient of as 1.

- We already have the coefficient of the quadratic as 1 therefore, therefore no need to divide the entire equation with the leading coefficient.

**Step3: **Find the value of coefficient “b” of new equation .Find the value of thereby.

- On comparision with the standard form ,the value of .Therefore the value of and value of = 9.

**Step4: **Add and subtract the term term.

**Step5:** Group the First term, Middle term and + term to complete the square and take the rest of the terms to the right side of “=” sign.

- Grouping the first, middle and + terms as

**Step6: **Write the Completed square as and solve for the variable “x” taking square root both sides.

- can be written as .Hence,taking square root both sides we get .Thererfore and the value of x would be.Hence the roots of the quadratic would be x= 8,-2.

Let us try a complex example.

**Strategy Step-by-Step:-**

**Step1**: First of all, replace y with 0.Identify the leading coefficient of the quadratic equation that is a.

- In the current problem, replacing y with 0, we have .On one to one comparison with the standard form the quadratic equation, we can say that the leading coefficient is a=4.

**Step2: **Divide the equation throughout by the leading coefficient “a” to make the coefficient of as 1.

- The leading coefficient “a” of the given quadratic is 4.So we divide the equation with 4 both sides.

**Step3: **Find the value of coefficient “b” of new equation .Find the value of thereby.

- On comparision with the standard form ,the value of .Therefore the value of and value of = .

**Step4: **Add and subtract the term term.

**Step5:** Group the First term, Middle term and + term to complete the square and take the rest of the terms to the right side of “=” sign.

- Groupingthe first, middle and + terms as = = (taking the common denomiantor as 16).

**Step6: **Write the Completed square as and solve for the variable “x” taking square root both sides.

- = can we writen as . Hence,taking square root both sides we get

Therefore .

and

Therefore x=

FInally,the simplified solutions would be x=