Question

The sum and the product of the zeroes of a quadratic polynomial are 3 and 9/4 respectively. Find the quadratic polynomial and its zeroes.

Answer 1

sum of zeroes of quadratic polynomial
$\alpha + \beta = 3 \: \: \: \: \: \: \: \: \: \: \:.......eq(1)$
and product of roots or zeroes is
$\alpha \times \beta = \frac{9}{4} \: \: \: \: \: \: \: \: \: \: \: \: ...........eq(2)$
here
$\beta = \frac{9}{4 \alpha }$
put value of beta in equation 1
$\alpha + \frac{9}{4 \alpha } = 3$
take LCM 4 alpha and solve
$4 { \alpha }^{2} + 9 = 12 \alpha$
$4 { \alpha }^{2} - 12 \alpha + 9 = 0$
solve this quadratic equation with splitting method
$4 { \alpha }^{2 } - 6 \alpha - 6 \alpha + 9 = 0$
after solving this we find value of alpha is
$(2 \alpha - 3)(2 \alpha - 3) = 0$
$\alpha = \frac{3}{2}$
put value of alpha is equation 1 or 2 and we can find value of beta also
$\frac{3}{2} + \beta = 3 \\ \beta = 3 - \frac{3}{2} \\ \beta = \frac{3}{2}$

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