To determine the number of real solutions of the quadratic equation \(4x^2 + 3x - 7 = 0\), we can use the discriminant \(D\), which is calculated using the formula:
\[ D = b^2 - 4ac \]
where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\). For this equation:
- \(a = 4\)
- \(b = 3\)
- \(c = -7\)
Now, substituting in these values:
\[ D = 3^2 - 4(4)(-7) = 9 - 4(4)(-7) \]
Calculating further:
\[ D = 9 - (-112) = 9 + 112 = 121 \]
Since the discriminant \(D = 121\) is positive, we can conclude that there are two real solutions to the equation.
The correct statement from the given options is:
There will be two real solutions since the discriminant is positive.