Algebra 2 Complex Conjugates and Complex Solutions Lesson (Free PDF)
This complex conjugates lesson for Algebra 2 covers complex conjugates and complex solutions (Common Core N-CN.A.3, N-CN.C.7), with a free PDF to print.
The conjugate of a + bi is a – bi, and a number times its conjugate equals a squared plus b squared. To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. A quadratic equation with real coefficients and a negative discriminant has two complex solutions that are conjugates.
Algebra 2 · Common Core State Standards: N-CN.A.3, N-CN.C.7
Key facts in this complex conjugates lesson
- The conjugate of a + bi is a – bi.
- z times its conjugate = a² + b², a real number.
- The modulus |a + bi| = √(a² + b²).
- Divide by multiplying top and bottom by the conjugate of the denominator.
- A negative discriminant gives a pair of complex conjugate solutions.
What is a complex conjugate?
A key fact: \(z \cdot \overline{z} = (a + bi)(a – bi) = a^{2} + b^{2}\). The product of a number and its conjugate is always a nonnegative real number. For \(3 + 2i\), the product is \(9 + 4 = 13\). Also, \(z + \overline{z} = 2a\) is real.
For \(3 + 4i\), the modulus is \(\sqrt{9 + 16} = 5\).
How do you divide complex numbers?
A fraction with \(i\) in the denominator is not in standard form. Multiply the numerator and denominator by the conjugate of the denominator. The new denominator becomes a real number.
- Write the conjugate of the denominator.
- Multiply the top and bottom by it.
- Simplify using \(i^{2} = -1\).
- Write the result as \(a + bi\).
Example: \(\frac{5 + 10i}{2 – i} = \frac{(5 + 10i)(2 + i)}{(2 – i)(2 + i)} = \frac{10 + 5i + 20i + 10i^{2}}{4 + 1} = \frac{25i}{5} = 5i\).
How do you solve a quadratic with complex solutions?
Use the quadratic formula. When the discriminant \(b^{2} – 4ac\) is negative, the square root is imaginary, and the solutions are a conjugate pair \(p \pm qi\). The graph of such a quadratic never touches the x-axis.
For \(x^{2} – 4x + 5 = 0\): \(x = \frac{4 \pm \sqrt{16 – 20}}{2} = \frac{4 \pm 2i}{2} = 2 \pm i\).
Complex conjugates lesson: questions and answers
What is the conjugate of a complex number?
The conjugate of a + bi is a – bi. Only the sign of the imaginary part changes. For example, the conjugate of 5 – 2i is 5 + 2i.
Why do you multiply by the conjugate when dividing?
A number times its conjugate equals a squared plus b squared, which is real. That removes i from the denominator, so the quotient can be written in a + bi form.
How do you find the modulus of a complex number?
Use the Pythagorean theorem: the modulus of a + bi is the square root of a squared plus b squared. For 5 – 12i, the modulus is the square root of 169, which is 13.
When does a quadratic equation have complex solutions?
When the discriminant b squared minus 4ac is negative. Then the parabola does not cross the x-axis and the two solutions are complex conjugates.
Do complex solutions always come in pairs?
For a polynomial with real coefficients, yes. If p + qi is a solution, then p – qi is also a solution.
How to use this complex conjugates lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the complex conjugates lesson as a PDF with the button below and keep it with your notes.
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