Question

Find a quadratic polynomial whose sum and product of its zeros is 1, -4 respectively?

Answer 1

Hint: In this question we have to form a quadratic equation whose zeros is given. Roots can also be termed as zeros .There are two ways to solve this problem. One is using the direct formula based approach to find the quadratic equation when some and product of roots are given and another one is using the condition given in the question formulate relation between the coefficients of any general quadratic equation $a{x^2} + bx + c = 0$.

Complete step-by-step answer:
As we know zeros are nothing but the roots of the polynomial.
And for a quadratic equation $a{x^2} + bx + c = 0$ the sum of the roots is the ratio of negative time’s coefficient of x to coefficient of ${x^2}$, and the product of roots is the ratio of constant term to coefficient of ${x^2}$.
Let the roots of this quadratic equation be (x1, x2).
$ \Rightarrow {x_1} + {x_2} = \dfrac{{ - b}}{a},{x_1}{x_2} = \dfrac{c}{a}$
Now it is given that the sum of the roots is 1
$ \Rightarrow {x_1} + {x_2} = 1 = \dfrac{{ - b}}{a}$
$ \Rightarrow b = - a$
And the product of the roots is -4
$ \Rightarrow {x_1}{x_2} = \dfrac{c}{a} = - 4$
$ \Rightarrow c = - 4a$
Now substitute the values of b and c in the above quadratic equation we have,
$ \Rightarrow a{x^2} + \left( { - a} \right)x + \left( { - 4a} \right) = 0$
Now divide by a throughout we have,
$ \Rightarrow {x^2} - x - 4 = 0$
So this is the required quadratic equation whose sum and product of zeros is 1 and -4.

Note: Let’s discuss the second approach, any quadratic equation can directly be formulated using the sum and product of roots. The direct formula for it is ${x^2} - \left( {{\text{sum of roots}}} \right)x + {\text{product of roots = 0}}$. This direct formula can be used when there is a time constraint to solve the problem as it helps to save time.