Use the discriminant to determine the nature of the roots for each equation.
a) 4x^2 – 5x – 4 = 0
To find the discriminant, we can use the formula D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
For the equation 4x^2 - 5x - 4 = 0, we have a = 4, b = -5, and c = -4. So, the discriminant is given by:
D = (-5)^2 - 4(4)(-4)
= 25 + 64
= 89
Since the discriminant is positive and greater than zero (D > 0), the equation has two distinct real roots.