Complex roots are the solutions of an equation that involve imaginary numbers and are usually written in the form a + ib, where i² = -1.
Example 1: Find the roots of the quadratic equation represented as x2 - 4x + 13 = 0.
Solution:
We have equation as x2-4x+13 = 0
Comparing with ax2+bx+c, we get, a = 1, b = -4 and c = 13
We know that, D = b2-4ac = (-4)2 - 4(1)(13) = 16 - 52 = -36
Now, we get √(-36) = i√(36) = 6i
Using the quadratic formula to find roots,
α, β = (-b ± √D)/2a
α, β = (-(-4) ± √(-36))/2 = (4 ± 6i)/2 = 2 ± 3i
Hence, the complex roots of the equation are 2 ± 3i
Example 2: Find the quadratic equation which has its roots as 3 ± 4i.
Solution:
Given that, roots of the quadratic equation are 3 ± 4i, we can write the equation as,
(x - (3+4i)*(x - (3-4i)) = 0
x2 - (3+4i)x - (3-4i)x + (3+4i)(3-4i) = 0
x2 - (3+4i+3-4i)x + (32 + 42) = 0
x2 - 6x + 25 = 0
Thus, we have obtained the quadratic equation as x2 - 6x + 25 = 0 whose roots are given by 3 ± 4i
Practice Problems
Problem 1: Find the roots of the equation: x3 - 1 = 0.
Problem 2: The roots of a quadratic equation are given as 4 ± 3i, find its equation.
Problem 3: Find the quadratic equation whose roots are +2i and -2i, where, i = √-1.
Problem 4: Find the roots of the quadratic equation represented as x2-3x+15=0.
Problem 5: Solve the equation: x2- 5x + 12=0.
