If one root of the equation x^2 - 30x + p = 0 | vedclass

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Question

(B) Let the roots of the equation $x^2 - 30x + p = 0$ be $\alpha$ and $\alpha^2$.From the relation between roots and coefficients:Sum of roots: $\alph

Vedclass · Accepted Answer

$125, -216$

Vedclass · Answer

(B) Let the roots of the equation $x^2 - 30x + p = 0$ be $\alpha$ and $\alpha^2$.From the relation between roots and coefficients:Sum of roots: $\alpha + \alpha^2 = 30$  $(1)$Product of roots: $\alpha \cdot \alpha^2 = \alpha^3 = p$  $(2)$From equation $(1)$,$\alpha^2 + \alpha - 30 = 0$.Factoring the quadratic: $(\alpha + 6)(\alpha - 5) = 0$.So,$\alpha = -6$ or $\alpha = 5$.If $\alpha = -6$,then $p = \alpha^3 = (-6)^3 = -216$.If $\alpha = 5$,then $p = \alpha^3 = (5)^3 = 125$.Therefore,the possible values for $p$ are $125$ and $-216$.

If one root of the equation $x^2 - 30x + p = 0$ is the square of the other root,then $p = \dots \dots \dots$

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