Question

If -5 is a root of the quadratic equation $2{x^2} + px - 15 = 0$ and the quadratic equation $p({x^2} + x) + k = 0 $has equal roots, find the value of k.

Answer 1

Hint – In this question let $\alpha $ be another root of the quadratic equation along with -5. Use the concept of sum and product of roots to find the value of $\alpha {\text{ and p}}$. Now for the second equation the roots are equal thus let the roots be $\beta ,\beta $. This will help getting the value of k.

Complete Step-by-Step solution:
Given quadratic equation is
$2{x^2} + px - 15 = 0$
Now it is given that one of the root of this quadratic equation is (-5)
So let us consider that the other root is $\alpha $.
Now as we know that in a quadratic equation the sum of roots is the ratio of negative time’s coefficient of x to the coefficient of x2.
$ \Rightarrow - 5 + \alpha = \dfrac{{ - \left( p \right)}}{2} = \dfrac{{ - p}}{2}$..................... (1)
Now we also know that in a quadratic equation the product of roots is the ratio of constant term to the coefficient of x2.
$ \Rightarrow - 5\alpha = \dfrac{{ - 15}}{2}$
Now simplify the above equation we have,
$ \Rightarrow \alpha = \dfrac{{15}}{{2 \times 5}} = \dfrac{3}{2}$
Now substitute this value in equation (1) we have,
$ \Rightarrow - 5 + \dfrac{3}{2} = \dfrac{{ - p}}{2}$
Now simplify the above equation we have,
$ \Rightarrow \dfrac{{ - 10 + 3}}{2} = \dfrac{{ - p}}{2}$
$ \Rightarrow \dfrac{{ - 7}}{2} = \dfrac{{ - p}}{2}$
$ \Rightarrow p = 7$
Now the other quadratic equation becomes
$ \Rightarrow 7\left( {{x^2} + x} \right) + k = 0$
$ \Rightarrow 7{x^2} + 7x + k = 0$
Now let the roots of this quadratic equation are $\beta ,\beta $ “(as equal roots are given).
Now the sum of roots is the ratio of negative time’s coefficient of x to the coefficient of x2.
$ \Rightarrow \beta + \beta = \dfrac{{ - 7}}{7} = - 1$
$ \Rightarrow 2\beta = - 1$
$ \Rightarrow \beta = \dfrac{{ - 1}}{2}$
Now the product of roots is the ratio of constant term to the coefficient of x2.
\[ \Rightarrow {\beta ^2} = \dfrac{k}{7}\]
Now substitute the value of ($\beta $) in above equation we have,
\[ \Rightarrow {\left( {\dfrac{{ - 1}}{2}} \right)^2} = \dfrac{k}{7}\]
$ \Rightarrow \dfrac{k}{7} = \dfrac{1}{4}$
$ \Rightarrow k = \dfrac{7}{4}$
So this is the required value of the k.
So this is the required answer.

Note – There can be another method to solve this problem, in the second approach the procedure till finding the value of p will remain the same however after that we can use the condition for equal roots that is ${b^2} - 4ac = 0$for any quadratic equation of the form $a{x^2} + bx + c = 0$. This too will give the same value of k as obtained above.