To find the discriminant of the quadratic equation \(4v^2 + 9v = 0\), we first want to express it in the standard form \(ax^2 + bx + c = 0\).
The given equation can be rewritten as:
\[ 4v^2 + 9v + 0 = 0 \]
Here, we identify:
- \( a = 4 \)
- \( b = 9 \)
- \( c = 0 \)
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = 9^2 - 4 \cdot 4 \cdot 0 \] \[ D = 81 - 0 \] \[ D = 81 \]
Now, we can analyze the discriminant to determine how many real solutions the equation has. The rules for the number of real solutions based on the discriminant are:
- If \( D > 0 \), there are 2 distinct real solutions.
- If \( D = 0 \), there is exactly 1 real solution (a double root).
- If \( D < 0 \), there are no real solutions.
Since \( D = 81 \), which is greater than 0, the equation \(4v^2 + 9v = 0\) has 2 distinct real solutions.