Write a quadratic equation with roots of 4 + 3i and 4 – 3i. Show all your work.
To write a quadratic equation with complex roots, we need to keep in mind that complex roots always come in conjugate pairs. So, if we know that the roots are 4 + 3i and 4 - 3i, we can conclude that the conjugate of 4 + 3i is 4 - 3i.
A quadratic equation in the form of ax^2 + bx + c = 0 can be determined by using the following rule:
1. The sum of the roots, (4 + 3i) + (4 - 3i), can be found by adding the real and imaginary parts of the numbers separately, giving 8 as the sum since the imaginary parts cancel each other out.
2. The product of the roots, (4 + 3i)(4 - 3i), can be determined by calculating (4 * 4) - (3i * 3i), which simplifies to 16 - 9i^2. Since i^2 is equal to -1, the equation simplifies further to 16 + 9, giving 25 as the product.
Now that we have the sum and product of the roots, we can write our quadratic equation:
x^2 - (Sum of roots) * x + Product of roots = 0
x^2 - 8x + 25 = 0
Therefore, the quadratic equation with roots of 4 + 3i and 4 - 3i is x^2 - 8x + 25 = 0.