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Question
Solve the following quadratic equation by completing the square method.
x2 + x – 20 = 0
Solution
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.
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