The equation of a curve is given by \(y = 2x^{2} - 5x + k\). If the curve has two intercepts on the x- axis, find the value(s) of constant k.
Answer Details
To find the x-intercepts of the curve, we need to set y = 0 and solve for x. Therefore, we have:
$$
\begin{aligned}
y &= 2x^{2} - 5x + k \\
0 &= 2x^{2} - 5x + k \\
\end{aligned}
$$
For the curve to have two x-intercepts, the discriminant of the quadratic equation must be greater than zero:
$$
\begin{aligned}
b^{2} - 4ac &> 0 \\
(-5)^{2} - 4(2)(k) &> 0 \\
25 - 8k &> 0 \\
-8k &> -25 \\
k &< \frac{25}{8} \\
\end{aligned}
$$
Therefore, the value of constant k must be less than \(\frac{25}{8}\). Thus, the answer is \(k < \frac{25}{8}\).