4х²-5х-7=0

To solve the quadratic equation \(4x^2 - 5x - 7 = 0\), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, the coefficients are: - \(a = 4\) - \(b = -5\) - \(c = -7\) Now, we will calculate the discriminant (\(b^2 - 4ac\)): \[ b^2 - 4ac = (-5)^2 - 4(4)(-7) = 25 + 112 = 137 \] Since the discriminant is positive, there will be two distinct real roots. Now, we substitute the values into the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{137}}{2 \cdot 4} \] \[ x = \frac{5 \pm \sqrt{137}}{8} \] Thus, the solutions for the equation \(4x^2 - 5x - 7 = 0\) are: \[ x_1 = \frac{5 + \sqrt{137}}{8} \] \[ x_2 = \frac{5 - \sqrt{137}}{8} \] These are the two values of \(x\) that satisfy the original equation.