Question

Find all complex solutions of x² - 3x + 4 = 0.

A) 1) 3 - i, 2) 3 + i
B) 1) -2, 2) 4
C) 1) -1 - i, 2) -1 + i
D) 1) 2 - i, 2) 2 + i

Answer 1

The quadratic equation x² - 3x + 4 = 0 has complex solutions which are found using the quadratic formula: (3 + i√7) / 2 and (3 - i√7) / 2. None of the provided options A, B, C, or D match these solutions.

Step-by-step explanation:

To find all complex solutions of the quadratic equation x² - 3x + 4 = 0, we need to apply the quadratic formula, which is x = −b ± √(b² − 4ac) / (2a) for an equation of the form ax² + bx + c = 0. In this equation, a = 1, b = −3, and c = 4.

Plugging these values into the formula, we get:

x = −3 ± √((−3)² − 4(1)(4)) / (2(1))
x = −3 ± √(9 − 16) / 2
x = −3 ± √(−7) / 2

The square root of −7 is an imaginary number, which we can denote as √−7 = i√7. Thus, the equation becomes:

x = (3 ± i√7) / 2

This means the complex solutions of the equation are:

1. x = (3 + i√7) / 2
2. x = (3 - i√7) / 2

Therefore, neither of the options A, B, C, or D presented by the student matches the correct complex solutions we just found.