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If α and β be the zeroes of the polynomial $a x^{2} + bx + c$ then the value of $α^{2} + β^{2} =$ ____.

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Detailed Solution

If α and β be the zeroes of the polynomial

a x^{2} + bx + c

then the value of

α^{2} + β^{2} = \frac{b^{2} - 2 ac}{a^{2}}

.
Given, α and β are zeroes of the polynomial

a x^{2} + bx + c

.
We know that, the relationship between zeroes and coefficient of quadratic polynomial is

Sum of Zeroes = - \frac{Coefficient of x}{Coefficient of x^{2}}

and

Product of Zeroes = \frac{Constant Term}{Coefficient of x^{2}}

.
Then, we get,

α + β = - \frac{b}{a}

and

αβ = \frac{c}{a}

… (1)
Now,

α + β = - \frac{b}{a}

.
On squaring both sides,

{(α + β)}^{2} = {(- \frac{b}{a})}^{2}

We know that,

{(a + b)}^{2} = a^{2} + b^{2} + 2 ab

. Then,

α^{2} + β^{2} + 2 αβ = {(\frac{b}{a})}^{2}

\Rightarrow α^{2} + β^{2} + 2 \frac{c}{a} = \frac{b^{2}}{a^{2}}

\Rightarrow α^{2} + β^{2} = \frac{b^{2}}{a^{2}} - \frac{2 c}{a}

\Rightarrow α^{2} + β^{2} = \frac{b^{2} - 2 ac}{a^{2}}

Hence,

α^{2} + β^{2} = \frac{b^{2} - 2 ac}{a^{2}}

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