Evaluate the discriminant and determine the number \( 4 x^{2}-5 x+7=0 \) Part: \( 0 / 2 \) P

To evaluate the discriminant of the quadratic equation \( 4x^2 - 5x + 7 = 0 \), we use the formula \( D = b^2 - 4ac \). Here, \( a = 4 \), \( b = -5 \), and \( c = 7 \). Plugging in these values gives us: \[ D = (-5)^2 - 4(4)(7) = 25 - 112 = -87. \] Since the discriminant is negative (\( D < 0 \)), there are no real solutions to the equation, which means the quadratic has two complex conjugate roots. When you come across a negative discriminant, remember it indicates the graph of your quadratic opens upwards and does not intersect the x-axis. Instead, it dips below and then rises again, revealing the existence of two beautiful, imaginary roots. So, if you're looking for real solutions, this one's a no-go!