##
**Completing
the Square Method**

In

**completing the square method**for solving a quadratic equation of the form ax^{2}+ bx + c = 0, we transpose the constant term c to the right hand side of the equation and then the left hand side is expressed as a perfect square expression.
We use the following steps for solving quadratic
equation of the form ax

^{2}+ bx + c = 0 by completing the square method:####
__Steps__**:**

__Steps__

1. Rewrite the equation so that the terms
containing x

^{2}and x (i.e. variable) are on the left side and the constant term is on the right side.
2. If the coefficient of x

^{2}is other than 1, then reduce the coefficient of x^{2}as 1 by dividing on both sides by the coefficient of x^{2}.
3. Add the square of half of the coefficient
of x on both sides of the equation to make the left hand side of the equation a
complete square.

4. Express the left hand side of the equation
as a perfect square of a binomial.

5. Use the square root property on both sides
and calculate the roots.

This process of Completing the Square Method
will be clear by the following worked out examples.

###
*Workout
Examples*

*Workout Examples*

*Example 1: Solve the following quadratic equations by completing the square method.*

*a.*

*x*^{2}– 2x – 8 =0

*b.*

*2x*^{2}+ 5x – 12 = 0

*c.*

*3x*^{2}– 5x + 2 = 0

*Solution:*

*a) Here,*

*x*^{2}– 2x – 8 = 0

*or, x*^{2}– 2x = 8

*or, x*^{2}– 2x + 1 = 8 + 1*[adding both sides by 1]*

*or, (x – 1)*^{2}= 9

*or, x – 1 = ±√9*

*or, x – 1 = ±3*

*or, x = 1 ± 3*

*Taking +ve sign,*

*x = 1 + 3 = 4*

*Taking –ve sign,*

*x = 1 – 3 =*

*–*

*2*

*∴*

*x = 4,*

*–2*

*Solution:*

*b) Here,*

*2x*^{2}+ 5x – 12 = 0

*or, 2x*^{2}+ 5x = 12

*or, (2x*^{2}+ 5x)/2 = 12/2*[dividing both sides by 2]*

*or, x*^{2}+5x/2 = 6

*or, x*^{2}+ 2.x.5/4 + (5/4)^{2}= 6 + (5/4)^{2}*[adding both sides by (5/4)*

^{2}]

*or, (x + 5/4)*^{2}= 6 + 25/16

*or, (x + 5/4)*^{2}= 121/16

*or, x + 5/4 = ±√(121/16)*

*or, x + 5/4 = ± 11/4*

*or, x =*

*–*

*5/4 ± 11/4*

*Taking +ve sign,*

*x =*

*–*

*5/4 + 11/4 = (*

*–*

*5 + 11)/4 = 6/4 = 3/2*

*Taking –ve sign,*

*x =*

*–*

*5/4*

*–*

*11/4 = (*

*–5 – 11)/4 = –16/4 = –4*

*∴*

*x = 3/2,*

*–4*

*Solution:*

*c) Here,*

*3x*^{2}– 5x + 2 = 0

*or, 3x*^{2}

*–*

*5x =*

*–*

*2*

*or, (3x*^{2}

*–*

*5x)/3 =*

*–*

*2/3**[dividing both sides by 3]*

*or, x*^{2}

*–*

*5x/3 =*

*–*

*2/3*

*or, x*^{2}

*–*

*2.x.5/6 + (5/6)*^{2}=

*–*

*2/3 + (5/6)*^{2}*[adding both sides by (5/6)*

^{2}]

*or, (x*

*–*

*5/6)*^{2}=

*–*

*2/3 + 25/36*

*or, (x*

*–*

*5/6)*^{2}= (

*–*

*24 + 25)/36*

*or, x*

*–*

*5/6 = ±√(1/36)*

*or, x*

*–*

*5/6 = ± 1/6*

*or, x = 5/6 ± 1/6*

*Taking +ve sign,*

*x =*

*5/6 + 1/6 = (5 + 1)/6 =*

*6*

*/6 =*

*1*

*Taking –ve sign,*

*x =*

*5/6*

*–*

*1/6 = (*

*5 –1)/6 = 4/6 = 2/3*

*∴*

*x = 1,*

*2/3*

*You can comment your questions or problems on solving quadratic equations by ‘completing the square method’ here.*

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