Solving Quadratic Equations by Completing the Square

Learn about solving quadratic equations by completing the square concept. An equation which contains more than one terms are squared but no higher power in terms has the syntax, ax²+bxy+cy² =0, where a represents the numerical coefficient of x², b represents the numerical coefficient of xy, and c represents the numerical coefficient of y²

Example: x²+2xy+y²

• Factorization method
• Completing the Square method
• Quadratic formula method

Completing the Square method

For some of the equation we can’t find common factors easily. To solve such a equation we use this method

Steps for solving quadratic equations by completing the square method :

If the quadratic equation in the form of ax²+bx+c = 0

Step 1: If ‘a`!=` 1’, divide both side by the value of ‘a’ (coefficient of x² is ‘a’).

Step 2: Write the given equation with the constant term on the right side.

Step 3: Find the half of the coefficient of x and take the square of the term finally add on both sides for completes the square.

Step 4: Simplify the right hand side and also write the left hand side as a square.

Step 5: Equate and solve.

Below are the example on solving quadratic equations by completing the square

Example 1: Factor the equation by complete the square method 2x² - x - 5 = 0

Solution:

Step 1: Divide by 2 on both side

`(2x^2)/2 -x/2 - 5/2` =0

`x^2-x/2= 5/2`

Step 2: Take the half of the coefficient of x and square it

`-1/2` is the coefficient of x and its half is `-1/4`

Square of `-1/4` is ---- > `1/16`

Step 3: Take left hand side and convert it to squared form, and simplify the right hand side.

`x^2-1/2x+1/16 = 5/2+1/16`

`x^2-1/2x+1/16 = 5/2 * 8/8+1/16` [To make common denominator]

`(x-1/4)^2 = 40/16 +1/16`

`(x-1/4)^2= 41/16`

Step 4: Take square root on both side,

`x-1/4 = +-sqrt(41/16)`

`x= 1/4+-sqrt (41/16)`

Hence the answer for the given problem is `1/4 +-sqrt (41/16)`