Question
What are the complex solutions of polynomial equation? x2 - 4x + 13 = 0
Answers
GPT 3.5
To find the complex solutions of the polynomial equation x^2 - 4x + 13 = 0, we can use the quadratic formula
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = -4, and c = 13. Plugging these values into the quadratic formula, we get:
x = (-(-4) ± √((-4)^2 - 4(1)(13))) / (2(1))
Simplifying further:
x = (4 ± √(16 - 52)) / 2
x = (4 ± √(-36)) / 2
Since we have a negative value under the square root, we know that there are complex solutions. The expression √(-36) can be expressed as √(36) ∙ √(-1), which further simplifies to 6i, where i is the imaginary unit (√(-1)).
Thus, the solutions are:
x = (4 + 6i) / 2
x = (4 - 6i) / 2
Simplifying:
x = 2 + 3i
x = 2 - 3i
Therefore, the complex solutions of the polynomial equation x^2 - 4x + 13 = 0 are x = 2 + 3i and x = 2 - 3i.
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = -4, and c = 13. Plugging these values into the quadratic formula, we get:
x = (-(-4) ± √((-4)^2 - 4(1)(13))) / (2(1))
Simplifying further:
x = (4 ± √(16 - 52)) / 2
x = (4 ± √(-36)) / 2
Since we have a negative value under the square root, we know that there are complex solutions. The expression √(-36) can be expressed as √(36) ∙ √(-1), which further simplifies to 6i, where i is the imaginary unit (√(-1)).
Thus, the solutions are:
x = (4 + 6i) / 2
x = (4 - 6i) / 2
Simplifying:
x = 2 + 3i
x = 2 - 3i
Therefore, the complex solutions of the polynomial equation x^2 - 4x + 13 = 0 are x = 2 + 3i and x = 2 - 3i.