Book Online Demo

Check Your IQ

Try Test

Courses

Offline Centres

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

see full answer

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

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}}

Watch 3-min video & get full concept clarity