Question

Find the value of ‘$k$’ for which the following quadratic equation has real roots:

$2x^2-5x-k=0$

Answer 1

Step 1:

Comparison of the coefficients:

Compare the coefficients of the given equation with the standard quadratic equation, $a{x}^2+bx+c=0$.

$\begin{array}{rcl}a& =& 2\\ b& =& -5\\ c& =& -k\end{array}$

Step 2:

Find the discriminant:

The discriminant $D$ of the given equation can be calculated as,

$\begin{array}{rcl}D& =& b^2-4ac\\ & =& {\left(-5\right)}^2-4\times 2\times \left(-k\right)\\ & =& 25+8k\end{array}$

Since the given equation has real roots, so the discriminant must be greater than or equal to 0.

$$\begin{gathered} D\ge 0 \\ \begin{array}{rcl}& \Rightarrow & 25+8k\ge 0\\ & \Rightarrow & 8k\ge -25\\ & \Rightarrow & k\ge -\frac{25}{8}\end{array} \end{gathered}$$

Hence, the value of $k$ is $k\ge -\frac{25}{8}$.