The value of p for which one root of the equa | vedclass

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(B) Let the roots of the quadratic equation ${x^2} - 30x + p = 0$ be $\alpha$ and ${\alpha ^2}$.According to the relationship between roots and coeffi

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$125$ and $-216$

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(B) Let the roots of the quadratic equation ${x^2} - 30x + p = 0$ be $\alpha$ and ${\alpha ^2}$.According to the relationship between roots and coefficients:Sum of roots: $\alpha + {\alpha ^2} = 30$Product of roots: $\alpha \cdot {\alpha ^2} = {\alpha ^3} = p$Solving the sum equation: ${\alpha ^2} + \alpha - 30 = 0$$(\alpha + 6)(\alpha - 5) = 0$This gives $\alpha = -6$ or $\alpha = 5$.If $\alpha = -6$,then $p = {\alpha ^3} = {(-6)^3} = -216$.If $\alpha = 5$,then $p = {\alpha ^3} = {(5)^3} = 125$.Thus,the values of $p$ are $125$ and $-216$.

The value of $p$ for which one root of the equation ${x^2} - 30x + p = 0$ is the square of the other,are

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