If one root is the square of the other root in the equation $x^{2} + p x + q = 0,$ mark the correct relationship in the following options.

p^{3}

- q (3p + 1) +

q^{2}

= 0

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p^{3}

- q (3p - 1) +

q^{2}

= 0

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p^{3}

+ q (3p - 1) +

q^{2}

= 0

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p^{3}

- q (3p - 1) -

q^{2}

= 0

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Solution

The correct option is B $p^{3}$ - q (3p - 1) + $q^{2}$ = 0
Take the roots as 2 & 4, then find out the values of p & q.
Then, p = -(2 + 4) = -6 and q = 2*4 = 8.
$x^{2} - 6 x + 8 = 0$
Now, check options by putting values of p and q in every equation.

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If one root is square of the other root of the equation $x^{2} + p x + q = 0$ , then the relation between p and q is

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The equation $x^{2} + p x - q = 0$ has one root as a square root of the other.
If $p^{3} + q^{2} = q (1 + k p)$ , find the value of k.

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