If one root of the equation ax^2 + bx + c = 0 | vedclass

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Question

(B) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $n\alpha$.According to the relationship between roots and coefficients

Vedclass · Accepted Answer

$n{b^2} = ac{(n + 1)^2}$

Vedclass · Answer

(B) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $n\alpha$.According to the relationship between roots and coefficients:Sum of roots: $\alpha + n\alpha = -\frac{b}{a} \implies \alpha(n + 1) = -\frac{b}{a} \implies \alpha = -\frac{b}{a(n + 1)}$ ... $(i)$Product of roots: $\alpha \cdot n\alpha = \frac{c}{a} \implies n\alpha^2 = \frac{c}{a} \implies \alpha^2 = \frac{c}{na}$ ... (ii)Substituting the value of $\alpha$ from equation $(i)$ into equation (ii):$\left(-\frac{b}{a(n + 1)}\right)^2 = \frac{c}{na}$$\frac{b^2}{a^2(n + 1)^2} = \frac{c}{na}$Multiplying both sides by $a^2(n + 1)^2$:$b^2 = \frac{c \cdot a^2(n + 1)^2}{na}$$b^2 = \frac{ac(n + 1)^2}{n}$$n{b^2} = ac{(n + 1)^2}$.

If one root of the equation $ax^2 + bx + c = 0$ is $n$ times the other root,then:

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