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

Hint: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.