Solving Quadratic Equations By Completing the Square

Solving Quadratic Equations By Completing the Square

We usually solve quadratic equations by factoring method. If the quadratic equation is not factorable, we can solve its roots by completing the square . In this hub I present several examples with their solutions.

Example One : Solve for X^2 - 8X + 3 = 0

Transpose 3 to the other side : X^2 - 8X = -3

Divide 8(coefficient of X) by 2 = 4 then square it 4^2 = 16

Add 16 to both side of the equation : X^2 -8X + 16 = -3 + 16

Factor the left side as a perfect square trinomial : (X - 4 )^2 = 13

Get the square root of both : X - 4 = +- SQRT (13)

X1 = +SQRT(13) + 4

X2 = -SQRT(13) + 4

Example Two : Solve for the roots of 2X^2 -6X + 5 = 0

Divide the whole equation by two . X^2 must have no other coefficient other than one.

X^2 - 3X + 5/2 =0

Transpose 5/2 to the right side : X^2 -3X = -5/2

Divide 3 (coefficient of X) by 2 = 3/2 then square it ( 3/2 )^2 = 9/4

Add 9/4 to both side of the equation : X^2 - 3X + 9/4 = -5/2 + 9/4

Factoring the left side (X – 3/2)^2 = (-10+9)/4

( X – 3/2)^2 = -1/4

Get the square root of both :

X - 3/2 = +- i/2

X1 = +i/2 + 3/2 = ( i + 3 )/2

X2 = ( -i + 3)/2

Example Number Three : Solve for the roots of 3X^2 -7X + 2 = 0

Start by dividing the whole equation by three :

X^2 - 7/3 X = -2/3

(7/3) (1/2) = 7/6 ( 7/6)^2 = 49/36

X^2 - 7/3X + 49/36 = -2/3 + 49/36

(X - 7/6)^2 = (-24 + 49)/36

(X – 7/6) ^2 = 25/36

Get the SQRT : X – 7/6 = +- 5/6

X1 = 5/6 + 7/6 = 12/6 = 2

X2 = -5/6 + 7/6 = 2/6 = 1/3