# 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**