Question

If one zero of the quadratic polynomial $p\left( x \right)=4{{x}^{2}}-8kx-9$ is negative of the other, find the value of $k$.

Answer 1

Hint: In this problem we need to calculate the value of $k$ whether the given quadratic polynomial has equal and opposite roots. For this we are going to use the relation between the roots of the quadratic equation and the coefficients of the quadratic equation. First, we will compare the given quadratic equation with the standard quadratic equation which is $a{{x}^{2}}+bx+c$ and write the values of $a$, $b$, $c$. Now we will assume one root of the equation as $\alpha $, then according to the problem statement another root will become $-\alpha $. Now we will calculate the sum of the roots and equate it to the value of $-\dfrac{b}{a}$. We will simplify the obtained equation by using basic mathematical operations to get the required result.

Complete step-by-step solution:
Given that, $p\left( x \right)=4{{x}^{2}}-8kx-9$.
Comparing the above given quadratic equation with the standard quadratic equation $a{{x}^{2}}+bx+c$, then we will get
$a=4$, $b=-8k$, $c=-9$.
Let us assume that the one root of the given quadratic equation is $\alpha $.
In the problem they have mentioned that one zero or root of the equation is negative of the other, then the second root will become $-\alpha $.
Now we have the two roots of the given quadratic equation as $\alpha $, $-\alpha $.
The sum of the roots of the given quadratic equation is given by
$\begin{align}
  & S=\alpha +\left( -\alpha \right) \\
 & \Rightarrow S=\alpha -\alpha \\
 & \Rightarrow S=0 \\
\end{align}$
We have the relation between sum of the roots of the quadratic equation to the coefficients of the quadratic equation as
$S=-\dfrac{b}{a}$
Substituting all the know values in the above equation, then we will get
$0=-\dfrac{8k}{4}$
Simplifying the above equation by using basic mathematical operations, then we will have
$\begin{align}
  & -8k=0\left( 4 \right) \\
 & \Rightarrow k=\dfrac{0}{-8} \\
 & \therefore k=0 \\
\end{align}$
Hence the value of $k$ which satisfies the given conditions is $0$.


Note: In this problem we have only used one of the relations between the roots of quadratic equation and the coefficients of the quadratic equation which is $\text{Sum of roots}=-\dfrac{\text{Coefficient of }x}{\text{Coefficient of }{{x}^{2}}}$. We also have another relation which is $\text{Product of roots}=\dfrac{\text{Constant}}{\text{Coefficient of }{{x}^{2}}}$. We can use any one of the equations or two equations according to the given conditions to solve this type of problem.