Question

$\text{If a and b are the zeroes of a polynomial}p{x}^2-5x+q,\text{then find the values of p}\text{and q, if a + b = ab = 10.}$

• A. $5\text{and}\frac{1}{2}$
• B. $5\text{and}2$
• C. $\frac{1}{2}\text{and}5$
• D. $10\text{and}1$

Answer 1

The correct option is C $\frac{1}{2}\text{and}5$

$\text{We know that, for a quadratic}\text{equation}a{x}^2+bx+c=0,\text{Sum of roots =}\alpha +\beta \text{and product of roots =}\alpha \beta \text{where}\alpha \text{and}\beta \text{are the roots of the}\text{equation.}$

$\text{Also,}\alpha +\beta =\frac{-b}{a}\text{and}\alpha \beta =\frac{c}{a}$

$\text{So,}a+b=\frac{5}{p}\text{and}ab=\frac{q}{p}.$

$\text{It is given that a + b = ab = 10}$

$\Rightarrow 10=\frac{5}{p}$
$\text{Hence,}p=\frac{1}{2}$

$\text{Also,}ab=\frac{q}{p}$
$\Rightarrow 10=\frac{q}{p}$
$\Rightarrow q=10p$
$\Rightarrow q=10\times \frac{1}{2}$
$\Rightarrow q=5$