To determine the number of real solutions for the quadratic equation \(11v^2 + 8v - 1 = 0\), we can use the discriminant. The discriminant \(D\) is calculated using the formula:
\[ D = b^2 - 4ac \]
where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).
From the equation \(11v^2 + 8v - 1 = 0\):
- \(a = 11\)
- \(b = 8\)
- \(c = -1\)
Now, we can calculate the discriminant:
\[ D = 8^2 - 4(11)(-1) \] \[ D = 64 + 44 \] \[ D = 108 \]
Now we analyze the value of the discriminant:
- If \(D > 0\), there are two distinct real solutions.
- If \(D = 0\), there is exactly one real solution.
- If \(D < 0\), there are no real solutions.
Since \(D = 108\) is greater than 0, the quadratic equation \(11v^2 + 8v - 1 = 0\) has two real solutions.