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The roots of the equation $4 x^{2} + 4 bx - (a^{2} - b^{2}) = 0$ are [[1]] and [[2]].

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answer is A - B 2, – A + B 2.

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Detailed Solution

The roots of the equation

4 x^{2} + 4 bx - (a^{2} - b^{2}) = 0

are

\frac{a - b}{2}

and

- \frac{a + b}{2}

.
Given quadratic equation is,

4 x^{2} + 4 bx - (a^{2} - b^{2}) = 0

Then,

4 x^{2} + 4 bx - (a^{2} - b^{2}) = 0

⟹ 4 x^{2} + 4 bx - a^{2} + b^{2} = 0

⟹ 4 x^{2} + 4 bx + b^{2} - a^{2} = 0

⟹ {(2 x + b)}^{2} - a^{2} = 0

⟹ (2 x + b - a) (2 x + b + a) = 0

\{x^{2} - y^{2} = (x - y) (x + y)\}

⟹ 2 x + b - a = 0 or 2 x + b + a = 0

∴ x = \frac{a - b}{2} or x = - \frac{a + b}{2}

Therefore, the roots of the equation

4 x^{2} + 4 bx - (a^{2} - b^{2}) = 0

are

\frac{a - b}{2}

and

- \frac{a + b}{2}

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