Find the roots of the quadratic equation $x^{2} - 2 a x + a^{2} - b^{2} - c^{2} = 0$ , where $a, b, c \in R$

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Solution

Consider the given equation.

$x^{2} - 2 a x + a^{2} - b^{2} - c^{2} = 0$ …….. (1)

Let $α, β$ are the roots of given equation.

$α + β = \frac{- B}{A} = 2 a$ ……. (2)

$α β = \frac{C}{A} = a^{2} - b^{2} - c^{2}$ …….. (3)

Since, $(α - β) = \sqrt{{(α + β)}^{2} - 4 α β}$

$(α - β) = \sqrt{{(2 a)}^{2} - 4 (a^{2} - b^{2} - c^{2})}$

$(α - β) = \sqrt{4 a^{2} - 4 a^{2} + 4 b^{2} + 4 c^{2}}$

$(α - β) = \sqrt{4 b^{2} + 4 c^{2}}$

$(α - β) = 2 \sqrt{b^{2} + c^{2}}$ ……. (4)

On adding equation $(2)$ and $(4)$ , we get

$2 α = 2 a + 2 \sqrt{b^{2} + c^{2}}$

$α = a + \sqrt{b^{2} + c^{2}}$

On putting the value of $α$ in equation $(2)$ , we get

$β = a - \sqrt{b^{2} + c^{2}}$

Hence, this is the answer.

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