Question

For the Quadratic equation, ${x^2} + x - 5 = 0$, write the product of the roots.

Answer 1

Hint:
Here, we will equate the given quadratic equation with the general quadratic equation, to find the coefficients of the terms in the given equation. Then by using the coefficients and the formula of the product of roots, we will find the required product of the roots.
Formula Used: Product of the roots of the quadratic equation which is of the form $a{x^2} + bx + c = 0$ is given by the formula Product of the roots $ = \dfrac{c}{a}$

Complete step by step solution:
We are given with a Quadratic equation ${x^2} + x - 5 = 0$
The given Quadratic equation is of the form $a{x^2} + bx + c = 0$.
Thus, the coefficient of ${x^2}$ is $a = 1$ , the coefficient of $x$ is $b = 1$ and the constant term is $c = - 5$.
Product of the roots of the quadratic equation which is of the form $a{x^2} + bx + c = 0$ is given by the formula $\dfrac{c}{a}$
Substituting $c = - 5$ and $a = 1$ in the formula, we get
Product of the roots $ = - \dfrac{5}{1}$
$ \Rightarrow $ Product of the roots $ = - 5$

Therefore, the product of the roots of the quadratic equation ${x^2} + x - 5 = 0$ is $ - 5$.

Note:
We also know that the quadratic equation is of the form ${x^2} - $(Sum of roots)$x + $Product of roots. From the given quadratic equation ${x^2} + x - 5 = 0$, we have the sum of the roots as $ - 1$ and the product of the roots as $ - 5$. We know that the Quadratic equation is defined as the polynomial equation with the highest degree as 2. The solutions of the quadratic equation are called the roots of the quadratic equation. The product of the roots is defined as the ratio of the constant term to the leading coefficient which is also the coefficient of ${x^2}$. Sum of the roots is defined as the ratio of the negation of the coefficient of $x$ to the leading coefficient which is also the coefficient of ${x^2}$.