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Question
Solve the following quadratic equation by completing the square method.
2y2 + 9y + 10 = 0
Solution
2y2 + 9y +10 = 0
`y^2 + 9/2y + 5 = 0` ...[Dividing both sides by 2]
If `y2 + 9/2y + k = (y + a)^2`, then
`y^2 + 9/2y + k = y^2 + 2ay + a^2`
Comparing the coefficients, we get
`9/2 = 2a and k = a^2`
∴ `a = 9/4 and k = (9/4)^2 = 81/16`
Now, `y^2 + 9/2y + 5 = 0`
∴ `y^2 + 9/2y + 81/16 - 81/16 + 5 = 0`
∴ `(y + 9/4)^2 + ((-81 + 80)/16) = 0`
∴ `(y + 9/4)^2 - 1/16 = 0`
∴ `(y + 9/4)^2 = 1/16`
Taking the square root of both sides, we get
`y + 9/4 = ± 1/4`
∴ `y + 9/4 = 1/4 or y + 9/4 = -1/4`
∴ `y = 1/4 - 9/4 or y = -1/4 - 9/4`
∴ `y = (-8)/4 = -2 or y = -10/4 = (-5)/2`
∴ The roots of the given quadratic equation are -2 and `(-5)/2`.
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