To find the discriminant of the quadratic equation \(4u^2 - 9u - 7 = 0\), we can use the formula for the discriminant \(D\) of a quadratic equation in the form \(au^2 + bu + c = 0\):
\[ D = b^2 - 4ac \]
Here, the coefficients are:
- \(a = 4\)
- \(b = -9\)
- \(c = -7\)
Now, we can substitute these values into the discriminant formula:
\[ D = (-9)^2 - 4(4)(-7) \]
Calculating each term:
- \( (-9)^2 = 81 \)
- \( 4 \times 4 \times -7 = -112\)
- Thus, \( D = 81 - (-112) = 81 + 112 = 193 \)
Now, \(D = 193\).
Since the discriminant \(D\) is greater than 0, \(D > 0\), this indicates that the quadratic equation has two distinct real solutions.