Question

If $ - a$ is a solution of equation ${x^2} + 3ax + k = 0$, find the value of $k$.

Answer 1

Hint: The given question is related to the concept of quadratic equation. \[a{x^2} + bx + c = 0\], where $a \ne 0$ is the standard form of a quadratic equation. The values of $x$ which satisfy the equation are called the roots or the solutions. These roots can be real or imaginary. Here in this question, we are already given a root i.e.,$ - a$. We just have to substitute $ - a$ in place of $x$ in order to get the value of $k$.

Complete step by step answer:
Here, we are given a quadratic equation ${x^2} + 3ax + k = 0$ and one of its roots as $ - a$, meaning one value of $x$. We have to find out the value of $k$ such that the left-hand side is equal to the right-hand side of the equation.
According to the question,
We have to substitute the value of $ - a$ in place of $x$. This is because it is given that $ - a$ is a solution of the equation. It means that for the value of $ - a$,$x$ is equal to zero.
Substituting $ - a$ in place of $x$ in the given quadratic equation ${x^2} + 3ax + k = 0$, we get,
$
   \Rightarrow {\left( { - a} \right)^2} + 3a\left( { - a} \right) + k = 0 \\
   \Rightarrow {a^2} - 3{a^2} + k = 0 \\
   \Rightarrow - 2{a^2} + k = 0 \\
   \Rightarrow k = 2{a^2} \\
 $
Therefore, the value of $k$ is $2{a^2}$.

Note:
As we have now found the value of $k$, there is a way to double-check our solution. In order to double-check the answer, all we have to do is substitute the value of $2{a^2}$ in place of $k$ in the given quadratic equation. If after substituting the value of $2{a^2}$ in place of $k$ and $ - a$ in place of $x$, we get zero i.e., LHS=RHS then we know for sure that our solution is correct.