(1+i)x^2+(3-i)x+2(1-3i)=0

4 min read Jun 16, 2024

Solving the Quadratic Equation: (1+i)x² + (3-i)x + 2(1-3i) = 0

This article will guide you through the process of solving the quadratic equation (1+i)x² + (3-i)x + 2(1-3i) = 0, where 'i' represents the imaginary unit (√-1).

Understanding the Problem

The given equation is a quadratic equation with complex coefficients. To solve it, we can use the quadratic formula, which is a general solution for any quadratic equation in the form ax² + bx + c = 0.

Applying the Quadratic Formula

The quadratic formula states:

x = (-b ± √(b² - 4ac)) / 2a

Where:

a = coefficient of x²
b = coefficient of x
c = constant term

In our case:

a = (1 + i)
b = (3 - i)
c = 2(1 - 3i)

Calculation Steps

Substitute the values: x = [-(3 - i) ± √((3 - i)² - 4(1 + i)(2(1 - 3i)))] / 2(1 + i)
Simplify the expression: x = [(-3 + i) ± √(9 - 6i + i² - 8(1 - 3i + i - 3i²))] / (2 + 2i) x = [(-3 + i) ± √(9 - 6i - 1 - 8 + 24i + 8i + 24)] / (2 + 2i) x = [(-3 + i) ± √(32 + 20i)] / (2 + 2i)
Find the square root of (32 + 20i): To find the square root of a complex number, we express it in polar form:
- Magnitude: √(32² + 20²) = √1284 ≈ 35.83
- Angle: tan⁻¹(20/32) ≈ 31.81°
Therefore, √(32 + 20i) ≈ 35.83(cos(31.81°) + i sin(31.81°)).
Substitute and simplify: x = [(-3 + i) ± 35.83(cos(31.81°) + i sin(31.81°))] / (2 + 2i) x = [(-3 + i) ± (35.83cos(31.81°) + 35.83i sin(31.81°))] / (2 + 2i)
Rationalize the denominator: Multiply the numerator and denominator by the conjugate of (2 + 2i), which is (2 - 2i): x = [(-3 + i) ± (35.83cos(31.81°) + 35.83i sin(31.81°))] * (2 - 2i) / [(2 + 2i)(2 - 2i)] x = [(-6 + 6i + 2i + 2 + 71.66cos(31.81°) - 71.66i sin(31.81°) + 71.66i sin(31.81°) + 71.66cos(31.81°))] / 8 x = [-4 + 8i + 143.32cos(31.81°)] / 8
Separate into real and imaginary parts: x = (-4 + 143.32cos(31.81°)) / 8 + (8/8)i x ≈ 16.58 + i

Solutions

Therefore, the solutions to the quadratic equation (1+i)x² + (3-i)x + 2(1-3i) = 0 are approximately:

x ≈ 16.58 + i
x ≈ -2.58 + i

Conclusion

We successfully solved the complex quadratic equation using the quadratic formula and complex number manipulation. The final solutions are complex numbers, indicating the roots of the equation lie in the complex plane.