Solutions of Quadratic Equations by Completing the Square


  • method of completing the square


In the previous section, you have learnt one method of obtaining the roots of a quadratic equation. In this section, we shall study another method. To implement this method we have certain steps to follow. We will understand the steps with `2x^2+7x-9=0` this example-
1) Make the coefficient of ` x^2` as 1. For that we have to divide the whole equation with the coefficient of `x^2`. In the above example coefficient of `x^2` is 2 so will divide the equation with 2, we get, `x^2+"7x"/2-9/2=0` 

2) After that take the constant to RHS. So we get `x^2+"7x"/2=9/2`.

3) Now take the square of the coefficient of x after multiplying with `1/2`, and then add that number to both the sides i.e add: `(1/2× "coefficient" "of"   x)^2`

Coefficient of x here is `7/2, (1/2×7/2)^2 = (7/4)^2`

by adding `(7/4)^2` both the sides we get, `x^2+"7x"/2+(7/4)^2 = 9/2+(7/4)^2`

4) Use `(a+b)^2` or `(a-b)^2`

`x^2+"7x"/2+(7/4)^2 = 9/2+(7/4)^2`

By obervation we get, `c= 9/2 + 49/16`

`(x+7/4)^2= (72+49)/16`

`(x+7/4)^2= 121/16`

`x+7/4 = + or - sqrt 121/16`


`x+7/4 = + or - 11/4`

`x+7/4 = 11/4 or -11/4`

`x=11/4-7/4 or -11/4-7/4`

`x= 4/4 or -18/4`

`x= 1 or -9/2`

Thus, the roots of `2x^2+7x-9=0`  are `1 or -9/2`

We can solve more examples using complete square method to understand easily.

If you would like to contribute notes or other learning material, please submit them using the button below. | Quadratic Equation part 7 (Solution by Completing Squares)

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