Complex numbers question ;)

In summary, the quadratic formula gives two solutions using one formula, with the plus or minus sign indicating the difference between the two solutions. In this case, the other root of the equation x^2 + ax + b = 0 is 4 - 5i. However, if a and b are not specified as real numbers, the other root could be any complex number.
  • #1

Homework Statement

One root of the equation x^2 + ax + b = 0 is 4 + 5i.

Write down the second root.

Homework Equations


The Attempt at a Solution

My problem is it's a "write down" question which suggests no working required. This is probably so simple but I just don't know... I cannot do this even thought I know the later parts of the question. Thank you in advance ;)
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  • #2
Write down the quadratic equation as you know it. The quadratic formula gives you two equations using one formula. What changes about the formula that gives you 2 separate solutions?
  • #3
The plus or minus, so would it be 4 - 5i?
  • #4
Yup! The quadratic formula gives solutions as

\frac{{ - b}}{{2a}} \pm \frac{{\sqrt {b^2 - 4ac} }}{{2a}}

However, you can rewrite the quadratic formula by pulling out a -1 from the term in the square root to get

\frac{{ - b}}{{2a}} \pm i\frac{{\sqrt {4ac - b^2 } }}{{2a}}

So you can identify the 4 and the 5 with the real and imaginary parts of the equation.
  • #5
Hehe, thank you so much :)

I know it was rather simple but my mind just wouldn't trigger.

Ty again ;)
  • #6
By the way, that answer is correct only if [itex]a[/itex] and [itex]b[/itex] are both real. If the problem doesn't specify that they are, then the other root could be any complex number.

1. What are complex numbers?

Complex numbers are numbers that contain both a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part with i representing the square root of -1.

2. How are complex numbers useful in science?

Complex numbers are used in a variety of scientific fields, including physics, engineering, and economics. They are especially useful in the study of electrical circuits, signal processing, and quantum mechanics.

3. What is the difference between a real number and a complex number?

A real number is a number that can be found on the number line, while a complex number cannot. Real numbers do not contain an imaginary part, whereas complex numbers do. Additionally, real numbers can be thought of as a subset of complex numbers.

4. How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply combine the real parts and the imaginary parts separately. For example, (3 + 2i) + (4 + 5i) = (3+4) + (2i+5i) = 7 + 7i. You can also think of it as adding or subtracting vectors in the complex plane.

5. What is the complex conjugate of a complex number?

The complex conjugate of a complex number a + bi is a - bi. It is found by changing the sign of the imaginary part. The product of a complex number and its complex conjugate always results in a real number, making it useful in simplifying complex expressions.

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