(a) To compute the discriminant, we can consider the equation in the form of ax^2 + bx + c = 0, where a = 4, b = -16, and c = 25.
The discriminant, denoted as Δ, is given by the formula: Δ = b^2 - 4ac.
Substituting the values, we have:
Δ = (-16)^2 - 4 * 4 * 25
= 256 - 400
= -144
Since the discriminant, Δ, is negative (-144), there are two different imaginary-number solutions to the equation.
(b) Now, let's use the quadratic formula to find the exact solutions of the equation:
The quadratic formula is given by: x = (-b ± √Δ) / (2a)
Substituting the values, we have:
x = (-(-16) ± √(-144)) / (2 * 4)
= (16 ± √144i) / 8
To simplify further, we can divide the numerator and denominator by 8:
= (2 ± √18i)
Therefore, the exact solutions of the equation 4x^2 – 16x + 25 = 0 are:
x = 2 + √18i
x = 2 - √18i