If the zero of the polynomial $x^{2} - p x + q$ are doubled in value to the zeroes of $2 x^{2} - 5 x - 3$ , find the value of $p$ and $q$ .

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Solution

$2 x^{2} - 5 x - 3 = 0$

$2 x^{2} - 6 x + x - 3 = 0$

$(x - 3) (2 x + 1) = 0$

$x = 3, - \frac{1}{2}$

Now,
Zeroes of the polynomial $x^{2} - p x + q$ are double in values to the zeroes of polynomial $2 x^{2} - 5 x - 3$ .
Therefore,
Zeroes of polynomial $x^{2} - p x + q$ will be- $6, - 1$
Therefore,
Sum of roots $= \frac{- b}{a}$
$6 + (- 1) = - (- p)$
$\Rightarrow p = 5$
Product of root $= \frac{c}{a}$
$6 \times - 1 = q$
$\Rightarrow q = - 6$
Hence the values of $p$ and $q$ are $5$ and $- 6$ respectively.

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