Question

If the sum and the product of the polynomial ${\mathrm{ax}}^2-5\mathrm{x}+\mathrm{c}$ is equal to $10$ each. Find the values of $a$ and $c$.

Answer 1

Find the coefficient of the polynomial:

Given polynomial is ${\mathrm{ax}}^2-5\mathrm{x}+\mathrm{c}$

We know that for a polynomial $p{x}^2+qx+r$ if $\alpha ,\beta$ are the roots of the equation then

1. $\alpha +\beta =-\frac{q}{p}$
2. $\alpha \beta =\frac{r}{p}$

Now comparing the given polynomial ${\mathrm{ax}}^2-5\mathrm{x}+\mathrm{c}$ with the standard form we have

1. $\alpha +\beta =\frac{5}{a}$
2. $\alpha \beta =\frac{c}{a}$

Given that the sum and the product of the root of the polynomial ${\mathrm{ax}}^2-5\mathrm{x}+\mathrm{c}$ is equal to $10$ each.

$$\begin{gathered} \therefore \alpha +\beta =10 \\ \Rightarrow \frac{5}{a}=10 \\ \Rightarrow a=\frac{1}{2} \end{gathered}$$

$$\begin{gathered} \therefore \alpha \beta =\frac{c}{a} \\ \Rightarrow 10=\frac{c}{{\displaystyle \frac{1}{2}}} \\ \Rightarrow c=5 \end{gathered}$$ $\left[\because a=\frac{1}{2}\right]$

Hence, $a=\frac{1}{2}$ and $c=5$