If the sum of the roots of the equation ax^2 | vedclass

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Question

(C) Let $\alpha$ and $\beta$ be the two roots of the quadratic equation $ax^2 + bx + c = 0$.According to the properties of roots,the sum of the roots

Vedclass · Accepted Answer

$b(a + b) = 2ac$

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(C) Let $\alpha$ and $\beta$ be the two roots of the quadratic equation $ax^2 + bx + c = 0$.According to the properties of roots,the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha\beta = \frac{c}{a}$.The sum of the squares of the roots is given by $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$.Substituting the values,we get $\alpha^2 + \beta^2 = (-\frac{b}{a})^2 - 2(\frac{c}{a}) = \frac{b^2}{a^2} - \frac{2c}{a} = \frac{b^2 - 2ac}{a^2}$.Given that the sum of the roots is equal to the sum of their squares,we have $\alpha + \beta = \alpha^2 + \beta^2$.Substituting the expressions,$-\frac{b}{a} = \frac{b^2 - 2ac}{a^2}$.Multiplying both sides by $a^2$,we get $-ab = b^2 - 2ac$.Rearranging the terms,$2ac = b^2 + ab$,which simplifies to $2ac = b(a + b)$.

If the sum of the roots of the equation $ax^2 + bx + c = 0$ is equal to the sum of their squares,then:

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