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Solution of Quadratic Equation by Completing Square

Geometry of Solution of Quadratic Equation by Completing Square

To solve a quadratic equation by the method of completing square is illustrated through the following examples.

Example

Solve the equation x2 − 3x − 4 = 0 by completing square.

Solution:

x2 − 3x − 4 = 0          (i)

Shifting constant term −4  to the right, we have

x2 − 3x = 4          (ii)

Adding the square of 1⁄2 × coefficient of x, that is,

(−3⁄2)2 on both sides of equation (ii), we get

x2 − 3x + (−3⁄2)2 = 4 + (−3⁄2)2          (ii)

As    x2 − 3x + (−3⁄2)2 = x2 − 1⁄2 × x × 3⁄2 + (−3⁄2)2 = (x−3⁄2)2, so we have

(x−3⁄2)2 = 4 + 9⁄4 = 4 + (16 + 9)/4

(x−3⁄2)2 = 25⁄4

Taking square root of both sides of the above equation,

√(x−3⁄2)2 = ±√(25⁄4)

⇒          x−3⁄2 = ± 5⁄2     or     x = 3⁄2 ± 5⁄2

Either     x = 3⁄2 + 5⁄2 = (3 + 5)/2 = 8⁄2 = 4     or     x = 3⁄2 − 5⁄2 = (3 – 5)/2 = −2⁄2 = −1

∴    4, −1 are the roots of the given equation.

Thus, the solution set is {−1, 4}.

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