Question

If the sum and the product of the polynomial $a{x^2} - 5x + c$ is equal to $10$ each , then find the values of $a$ and $c$.

Answer 1

Hint: The degree of a quadratic polynomial is $2$ , therefore it has two zeros or roots. The standard or general form of a quadratic equation is , $a{x^2} + bx + c = 0$ , where $a$, $b$, $c$ are coefficients of the equation. Here $a$ is the coefficient of ${x^2}$, $b$ is the coefficient of $x$ and $c$ is the constant term, where $a,b,c \in R$ .

Complete step by step answer:
Values of the coefficients according to the given question; $a = a$ , $b = - 5$ and $c = 1$ ;
Sum of roots $ = $ Product of roots $ = $ $10$ (given in the question)
And we are asked to find the values of $a$ and $c$ ;
To find the values of $a$ and $c$ , we must know the formula used for calculation of Sum of roots and product of roots. The formulae are listed as under;
 $\left( 1 \right){\text{Sum of roots}} = - \left( {\dfrac{{{\text{coefficient of }}x}}{{{\text{coefficient of }}{x^2}}}} \right)$
By putting the values of the coefficients, according to the given question;
$ \Rightarrow {\text{Sum of roots}} = - \dfrac{b}{a}$
Put the given value for sum of roots and the value of $b$ , we get;
$ \Rightarrow 10 = - \left( { - \dfrac{5}{a}} \right)$
$ \Rightarrow 10 = \dfrac{5}{a}$
Solving the above equation for $a$, we get ;
$ \Rightarrow a = \dfrac{1}{2}$ $......\left( 1 \right)$
$\left( 2 \right){\text{Product of roots}} = \left( {\dfrac{{{\text{constant}}}}{{{\text{coefficient of }}{x^2}}}} \right)$
$ \Rightarrow {\text{Product of roots}} = \dfrac{c}{a}$
$ \Rightarrow 10 = \dfrac{c}{a}$
Put the value of $a = \dfrac{1}{2}$ from equation $\left( 1 \right)$ , we get;
$ \Rightarrow 10 = 2c$
Therefore, $c = 5$
Therefore the value of $c$ is $5$ .
Therefore, the correct answer for this question is , the value of $a = \dfrac{1}{2}$ and $c = 5$ .
In the given question, the values of coefficients of the quadratic equation were asked, by putting the values of the respective coefficients i.e. $a$ and $c$ , then the quadratic equation will be;
$ \Rightarrow a{x^2} + bx + c = 0$
$ \Rightarrow \dfrac{1}{2}{x^2} - 5x + 5 = 0$
Simplifying further, we get the final quadratic equation as;
$ \Rightarrow {x^2} - 10x + 10 = 0$ $......\left( 2 \right)$
Using the standard quadratic equation formula i.e. $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ , roots of the equation (let ${x_1}$ and ${x_2}$ ) can be find out. The root values will be; ${x_1} = - 1.13$ and ${x_2} = - 8.87$ .

Note: The roots of a quadratic equation are those two values, such that by putting these values in the equation , our equation reduces to zero. That is why the roots of a quadratic equation are also called the zeroes of a quadratic equation. We can examine this fact by putting the values ${x_1} = - 1.13$ and ${x_2} = - 8.87$ in the equation $\left( 2 \right)$.