Question

The arithmetic mean of two numbers $a$ and $b$ is $9$ and the product is $- 5$. Write the quadratic equation whose roots are $a$ and $b$.

Answer 1

Quadratic equations are the polynomial equations of degree 2 in one variable of type $f\left( x \right){\text{ }} = a{x^2}\; + {\text{ }}bx{\text{ }} + {\text{ }}c{\text{ }} = {\text{ }}0$ where $a,{\text{ }}b,{\text{ }}c, \in R$ and $a \ne 0$. The values of variables satisfying the given quadratic equation are called its roots.

Complete step by step answer:
According to this question, the arithmetic mean of two numbers is $9$. Arithmetic mean (AP) or called average is the ratio of all observations to the total number of observations.So here in this question $AP = 9$. That is $\dfrac{{a + b}}{2} = 9$.
So, $a + b = 18$
Also, according to the question $a \times b = - 5$.
We also know that in any quadratic equation there is a polynomial equation of degree 2 or an equation that is in the form $a{x^2}\; + {\text{ }}bx{\text{ }} + {\text{ }}c{\text{ }} = {\text{ }}0$.
And also, equation should be made like this = $K{\text{ }}\left( {{x^2} - {\text{ }}sum{\text{ }}of{\text{ }}roots\left( x \right) + product{\text{ }}of{\text{ }}roots} \right)$
So, going by this formula the quadratic equation formed of roots $a$ and $b$ would be: $\Rightarrow K\left( {{x^2} - 18x + \left( { - 5} \right)} \right)\,$
$\therefore K\left( {{x^2} - 18x - 5} \right)$
where $K$ is an integer.

Hence, the quadratic equation whose roots are $a$ and $b$ is $K\left( {{x^2} - 18x - 5} \right)$.

Note: A quadratic equation is said to be any polynomial equation of degree 2 or an equation that is in the form $a{x^2}\; + {\text{ }}bx{\text{ }} + {\text{ }}c{\text{ }} = {\text{ }}0$. The quadratic formula, on the other hand, is a formula that is used for solving the quadratic equation. The formula is used to determine the roots/solutions to the equation.