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

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(B) Let the roots of the equation $ax^2 + bx + c = 0$ be $\alpha$ and $\alpha^2$.From the relation between roots and coefficients:Sum of roots: $\alph

Vedclass · Accepted Answer

$(a - b)^3$

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(B) Let the roots of the equation $ax^2 + bx + c = 0$ be $\alpha$ and $\alpha^2$.From the relation between roots and coefficients:Sum of roots: $\alpha + \alpha^2 = -b/a$Product of roots: $\alpha \cdot \alpha^2 = \alpha^3 = c/a$From the sum of roots,$\alpha^2 + \alpha + b/a = 0$. Cubing both sides or using the identity $(\alpha^2 + \alpha)^3 = (-b/a)^3$:$\alpha^6 + \alpha^3 + 3\alpha^3(\alpha^2 + \alpha) = -b^3/a^3$Substituting $\alpha^3 = c/a$ and $\alpha^2 + \alpha = -b/a$:$(c/a)^2 + (c/a) + 3(c/a)(-b/a) = -b^3/a^3$$c^2/a^2 + c/a - 3bc/a^2 = -b^3/a^3$Multiplying by $a^3$: $ac^2 + a^2c - 3abc = -b^3$$b^3 + a^2c + ac^2 = 3abc$.This condition is equivalent to $a(c - b)^3 = c(a - b)^3$.Thus,$X = (a - b)^3$.

If one root of the equation $ax^2 + bx + c = 0$ is the square of the other,then $a(c - b)^3 = cX$,where $X$ is

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