First slide

Theory of equations

Question

If the sum of the roots of the equation $x^{2} + px + q = 0$ is equal to the sum of their squares, then

Easy

A

$p^{2} - q^{2} = 0$
B

$p^{2} + q^{2} = 2 q$
C

$p^{2} + p = 2 q$
D

None of these

Solution

Let the roots be $α and β \Rightarrow α + β = - p, αβ = q$
Given, $α + β = α^{2} + β^{2}$
But $α + β = {(α + β)}^{2} - 2 αβ \Rightarrow - p = {(- p)}^{2} - 2 q$
$\Rightarrow p^{2} - 2 q = - p \Rightarrow p^{2} + p = 2 q$

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