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Among the subsequent options, the roots of the equation $x 2 + 6 x − a 2 + 2 a − 8 = 0$ by applying the quadratic formula are

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a − 2, − (a + 4)

a + 2, (a + 4)

a − 2, (a − 4)

a + 2, − (a − 4)

answer is A.

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Detailed Solution

The given quadratic equation is

x 2 + 6 x − a 2 + 2 a − 8 = 0 … … (1)

.
We have to find the roots of the equation.
Compare equation (1) with the standard form

a x 2 + b x + c = 0.

a = 1, b = 6, c = − a 2 + 2 a − 8

Find the determinant D by using the formula,

D = b 2 − 4 a c

.
Substitute the known values in the formula.

D = b 2 − 4 a c

D = 6^{2} - 4 \times 1 \times (- (a^{2} + 2 a - 8))

D = 36 + 4 a 2 + 8 a − 32

D = 4 a 2 + 8 a + 4 D = 4 (a 2 + 2 a + 1) D = 4 (a + 1) 2

The formula gives the roots of the quadratic equation,

− b ± D 2 a

.
Now,

D = 4 a + 1 2 D = 2 a + 1

Substitute the known values to seek out the roots of the equation.
.

α, β = − b + D 2 a, − b − D 2 a

α, β = − 6 + 2 (a + 1) 2 × 1, − 6 − 2 (a + 1) 2 × 1

α, β = − 6 + 2 a + 2 2, − 6 − 2 a − 2 2

α, β = a − 2, − (a + 4)

Hence, the roots of the equation are

a − 2, − (a + 4)

.
Therefore, option 1) is correct.

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