Question

If -5 is one of the root of quadratic equation $2{x^2} + px - 15 = 0$ and roots of quadratic equation $p\left( {{x^2} + x} \right) + k = 0$ is equal, then the value of $k$?

Answer 1

Hint: With the help of product and sum of roots, find the value of p in the equation $2{x^2} + px - 15 = 0$. In the similar way, find the roots of the equation $p\left( {{x^2} + x} \right) + k = 0$, to find the value of $k$.

Complete step-by-step answer:
Before going into the solution, let us all be clear about the concept.
If the roots of the quadratic equation $a{x^2} + bx + c = 0$ are $\alpha $ and $\beta $, the sum of the roots of the equation is \[\alpha + \beta = - \dfrac{b}{a}\] and product of the roots of the equation is \[\alpha \times \beta = \dfrac{c}{a}\]
So, in the equation $2{x^2} + px - 15 = 0$, one of the roots of the equation is given, let \[\beta = - 5\]
Then product of the roots of the equation $2{x^2} + px - 15 = 0$is,
$
   \Rightarrow \alpha \times \beta = \dfrac{c}{a} \\
   \Rightarrow \alpha \times \left( { - 5} \right) = \dfrac{{ - 15}}{2} \\
   \Rightarrow \alpha = \dfrac{3}{2}......\left( 1 \right) \\
 $
Sum of the roots of the equation $2{x^2} + px - 15 = 0$is,
$
   \Rightarrow \alpha + \beta = - \dfrac{b}{a} \\
   \Rightarrow \dfrac{3}{2} - 5 = - \dfrac{p}{2} \\
   \Rightarrow \dfrac{p}{2} = 5 - \dfrac{3}{2} \\
   \Rightarrow \dfrac{p}{2} = \dfrac{7}{2} \\
   \Rightarrow p = 7........\left( 2 \right) \\
 $
So, we got the value of p, which will be helpful in the equation $p\left( {{x^2} + x} \right) + k = 0$ to find the value of K.
Given that the roots of the equation $p\left( {{x^2} + x} \right) + k = 0$ are same, let the roots be ${\alpha _1}\,and\,{\beta _1}$
So, we have the equation $7{x^2} + 7x + k = 0$, where ${\alpha _1} = {\beta _1}$
So, sum of the roots of the equation $7{x^2} + 7x + k = 0$is,
\[
   \Rightarrow {\alpha _1} + {\beta _1} = - \dfrac{b}{a} \\
   \Rightarrow 2{\alpha _1} = - \dfrac{7}{7} \\
   \Rightarrow 2{\alpha _1} = - 1 \\
   \Rightarrow {\alpha _1} = \dfrac{{ - 1}}{2}.......\left( 3 \right) \\
 \]
Now find the product of the roots of the equation $7{x^2} + 7x + k = 0$, with the help of (3)
$
   \Rightarrow {\alpha _1} \times {\beta _1} = \dfrac{c}{a} \\
   \Rightarrow {\alpha _1}^2 = \dfrac{k}{7} \\
   \Rightarrow {\left( {\dfrac{{ - 1}}{2}} \right)^2} = \dfrac{k}{7} \\
   \Rightarrow k = \dfrac{7}{4} \\
 $
So, the value of k is $\dfrac{7}{4}$.

Note: There are alternative methods for this but solving through sum and product of the root formula, it becomes easy. Understand the condition provided and apply, compare the values of a, b and c, while substitutions.