Question

Find a quadratic polynomial, the sum and product of whose zeroes are −3 and 2 respectively.

Answer 1

Hint: Here we go through by assuming the general quadratic equation $a{x^2} + bx + c = 0$ and let their zeroes as $\alpha $ and $\beta $. Then apply the properties of quadratic equation i.e. $\alpha + \beta = - \dfrac{b}{a}$ and $\alpha \beta = \dfrac{c}{a}$ by using these properties we get the result.

Complete step-by-step answer:
Let the quadratic polynomial be $a{x^2} + bx + c = 0$ and its zeroes be $\alpha $ and$\beta $.
And we know that by the properties of the quadratic equation the sum of zeroes is equal to the negative of coefficient of x divided by coefficient of ${x^2}$. i.e. $\alpha + \beta = - \dfrac{b}{a}$
And the product of the zeroes is equal to the constant term divided by the coefficient of ${x^2}$. i.e. $\alpha \beta = \dfrac{c}{a}$.
Now in the question it is given that the sum of the zeroes are -3 that we can write as $\alpha + \beta = - \dfrac{b}{a} = - 3$… (1)
And the product of the zeroes is given as 2 that we can write as $\alpha \beta = \dfrac{c}{a} = 2$…. (2)
And we know that the general quadratic equation can be written as ${x^2}$- (sum of zeroes)x + product of zeroes.
By equation (1) and equation (2) we can clearly identify the sum of zeroes and the product of zeroes.
Now put these values in the general form of quadratic equation we get it as,
 ${x^2} + 3x + 2 = 0$
So, one quadratic polynomial will be ${x^2} + 3x + 2 = 0$.

Note: Whenever we face such a type of question the key concept for solving the question is. Always assume the quadratic equation in its general form and let its zeroes as $\alpha $ and $\beta $ then go according to the question and equate it to get the values of sum of roots and product of roots and then put the values of these in general form to get the required quadratic equation. And always take care of signs. We generally make mistakes here.