Final answer:
The roots of the quadratic equation 16x^2 − 64x + 65 = 0 are found using the quadratic formula, yielding the complex solutions − 2 + ¼ i and − 2 − ¼ i.
Step-by-step explanation:
To find the roots of the quadratic equation 16x2 − 64x + 65 = 0, we can use the quadratic formula, which states that the solutions for an equation of the form ax2 + bx + c = 0 are given by:
x = −b ± √(b2 − 4ac) / (2a)
In this case, a = 16, b = −64, and c = 65. Substituting these values into the formula, we get:
x = −64 ± √((-64)2 − 4(16)(65)) / (2 × 16)
x = −64 ± √(4096 − 4160) / 32
x = −64 ± √(−64) / 32
Since the discriminant (b2 − 4ac) is negative, this indicates that the equation has complex roots. So we can continue:
x = −64 ± √64i / 32
x = −64 ± 8i / 32
x = − 2 ± ¼i
Therefore, the roots of the equation in a + bi form are − 2 + ¼ i and − 2 − ¼ i.