(3+2i)x^2+(8+5i)x-3(1+i)=0

Jasonbradley

4 min read

·

Jun 16, 2024

26

Share

Solving the Complex Quadratic Equation: (3+2i)x²+(8+5i)x-3(1+i)=0

This article will guide you through the process of solving the complex quadratic equation (3+2i)x²+(8+5i)x-3(1+i)=0.

Understanding Complex Quadratics

A complex quadratic equation is a polynomial equation of degree two where the coefficients can be complex numbers (numbers that involve the imaginary unit 'i', where i² = -1). These equations can be solved using similar methods to those used for real quadratic equations, but with some adjustments to handle the complex coefficients.

Solving the Equation

1. The Quadratic Formula:

The most common method to solve quadratic equations is the quadratic formula. It states that the solutions to the equation ax² + bx + c = 0 are given by:

x = (-b ± √(b² - 4ac)) / 2a

2. Applying the Formula:

Let's apply the quadratic formula to our equation:

• a = 3 + 2i
• b = 8 + 5i
• c = -3(1 + i) = -3 - 3i

Now, substitute these values into the quadratic formula:

x = (-(8 + 5i) ± √((8 + 5i)² - 4(3 + 2i)(-3 - 3i))) / 2(3 + 2i)

3. Simplifying:

Simplify the equation step by step, paying close attention to the calculations involving complex numbers:

• Simplify the discriminant (b² - 4ac):

  • (8 + 5i)² = 39 + 80i
  • 4(3 + 2i)(-3 - 3i) = 48 + 12i
  • Discriminant = 39 + 80i - (48 + 12i) = -9 + 68i

• Calculate the square root of the discriminant:

  • √(-9 + 68i) = 8 + i (we can find this by using the polar form of complex numbers)

• Substitute back into the quadratic formula:

  • x = (-8 - 5i ± (8 + i)) / (6 + 4i)

4. Rationalizing the Denominator:

To get a simplified answer, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:

• x = ((-8 - 5i ± (8 + i)) * (6 - 4i)) / ((6 + 4i) * (6 - 4i))

5. Completing the Calculations:

Now, perform the multiplications and simplify the expression:

• x = ( (-48 - 20i + 48 + 8i) ± (48 + 8i - 24i - 4)) / (52)

• x = (-12i ± (44 - 16i)) / 52

• Therefore, the two solutions are:

  • x1 = (44 - 28i) / 52 = (11 - 7i) / 13
  • x2 = (-16 - 4i) / 52 = (-4 - i) / 13

Conclusion

By using the quadratic formula and performing careful calculations with complex numbers, we have successfully solved the complex quadratic equation (3+2i)x²+(8+5i)x-3(1+i)=0. The solutions are x = (11 - 7i) / 13 and x = (-4 - i) / 13. Remember to always pay close attention to the rules of complex number arithmetic and simplification when working with complex quadratic equations.