Question

Values of k for which the quadratic equation $ 2{{x}^{2}}+kx+k=0 $ has equal roots
A. 4,8
B. 0,4
C. 4,-8
D. 0,8

Answer 1

Hint: To solve this question we will use the quadratic formula method to solve the quadratic equation of the form $ a{{x}^{2}}+bx+c=0 $. The quadratic formula is $ x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} $ . In this formula, the part under the square root is known as the discriminant and it is denoted as D. When we have value $ D=0 $ it implies that the equation has two real and equal roots. So we use this condition to get the desired answer.

Complete step by step answer:
We have been given a quadratic equation $ 2{{x}^{2}}+kx+k=0 $ .
We have to find the value of k for the quadratic equation has equal roots.
Now, we know that the given equation is of the form $ a{{x}^{2}}+bx+c=0 $ and to have equal roots the value of discriminant of an equation will be equal to zero.
So, we have given the question that equation $ 2{{x}^{2}}+kx+k=0 $ has equal roots.
So, we have $ D=0 $
Now, we know that $ D=\sqrt{{{b}^{2}}-4ac} $
Now, substituting the values we get
 $ \begin{align}
  & \Rightarrow \sqrt{{{b}^{2}}-4ac}=0 \\
 & \Rightarrow \sqrt{{{k}^{2}}-4\times 2\times k}=0 \\
 & \Rightarrow \sqrt{{{k}^{2}}-8\times k}=0 \\
 & \Rightarrow {{k}^{2}}-8k=0 \\
\end{align} $
Now, simplifying further we get
 $ \Rightarrow k\left( k-8 \right)=0 $
Or we can write $ k=0 $ and $ k-8=0 $
So, we get two values of k $ k=0 $ and $ k=8 $ .
Option D is the correct answer.

Note:
 The number of roots of a quadratic equation depends on the degree of the equation. Here in this question, the degree of the equation is two; it means the equation has two roots. The roots of the quadratic equation can be found using one of the methods from the factorization method, the quadratic formula method, or completing the square method. Alternatively when $ D=0 $ the value of roots is determined by $ x=\dfrac{-b}{2a} $ .