Question

Solve the following quadratic equation by completing the square method.

x² + x – 20 = 0

Answer 1

x² + x – 20 = 0  

If x² + x + k = (x + a)², then x² + x + k = x² + 2ax + a²

Comparing the coefficients, we get,

∴ 1 = 2a and k = a²

∴ `1/2` = a and k = a² = `(1/2)^2 = 1/4`

Now, x² + x – 20 = 0 

`∴ x^2 + x + 1/4 - 1/4 - 20 = 0`

`∴ (x + 1/2)^2 - ((1 + 80)/4) = 0`

`∴ (x + 1/2)^2 - 81/4 = 0`

`∴ (x + 1/2)^2 = 81/4`

Taking square root of both sides, we get,

`∴ x + 1/2 = +- 9/2`

`∴ x + 1/2 = 9/2    "or"     x + 1/2 = - 9/2`

`∴ x = 9/2 - 1/2    "or"     x = - 9/2 - 1/2`

`∴ x = 8/2 = 4    "or"     x = - 10/2 = -5`

∴ The roots of the given quadratic equation are 4 and −5.