Question

The value of k for which the quadratic equation \[2{x^2} + kx + 2 = 0\] has equal roots is
a) 4
b) \[ \pm 4\]
c) \[ - 4\]
d) 0

Answer 1

Hint: Here in this question, we have to find the value of k, the equation is of the form of the quadratic equation. Here they have mentioned the root of the equation is equal. Therefore, the discriminant is given by \[{b^2} - 4ac = 0\] considering this formula and by simplifying we are going to obtain the value of k.

Complete step by step solution:
The equation is of the form of a quadratic equation. In general the quadratic equation will be in the form of \[a{x^2} + bx + c\]. The discriminant is the part of the quadratic formula underneath the square root symbol: \[{b^2} - 4ac\]. The discriminant tells us whether there are two solutions, one solution, or no solutions.
If the roots are equal the discriminant is equal to zero. If the roots are positive real the discriminant is greater than zero. If the roots are negative the discriminant is less than zero.
Consider the equation \[2{x^2} + kx + 2 = 0\]
The discriminant is a zero therefore we have \[{b^2} - 4ac = 0\]
Here the value of a is 2 and the value of b is k and the value of c is 2.
\[ \Rightarrow {(k)^2} - 4(2)(2) = 0\]
On simplifying we get
\[ \Rightarrow {k^2} - 16 = 0\]
Take 16 RHS we get
\[ \Rightarrow {k^2} = 16\]
Taking the square root on both sides we have
\[ \Rightarrow k = \pm 4\]
Therefore the value of k is equal to \[ \pm 4\]
Hence option B is the correct one.
So, the correct answer is “Option B”.

Note: The roots depend on the value of the discriminant. The discriminant is given by \[{b^2} - 4ac\]. The discriminant tells us whether there are two solutions, one solution, or no solutions. Suppose in the question if they mention the roots are real and distinct, then the discriminant will be \[{b^2} - 4ac > 0\]. If the roots are imaginary then the discriminant will be \[{b^2} - 4ac < 0\].