1. More with Complex Numbers Sec. 2.5b

2. Definition: Complex Conjugate The complex conjugate of the complex number is What happens when we multiply a complex number by its conjugate??? This is a positive real number!!!

3. Practice Problems Write the given complex numbers in standard form.

4. Complex Solutions of Quadratic Equations Remind me of the quadratic formula!!! What’s this part called? It can be used to tell whether the solutions to a particular quadratic equation are real numbers…  The discriminant!!!

5. Discriminant of a Quadratic Equation For a quadratic equation, where a, b, and c are real numbers and, If, there are two distinct real solutions. If, there is one repeated real solution. If, there is a complex conjugate pair of solutions.

6. Practice Problems Solve a = b = c = 1  Use the quadratic formula! A complex conjugate pair

7. Guided Practice Write the given complex number in standard form.

8. Guided Practice Write the given expression in standard form.

9. Guided Practice Write the given expression in standard form.

10. The Complex Plane Imaginary Axis Real Axis Imaginary Axis Real Axis

11. The Complex Plane Plot u = 1 + 3i, v = 2 – i, and u + v in the complex plane. Imaginary Axis Real Axis Notice that the two complex numbers, their sum, and the origin form a quadrilateral (what type?)  A Parallelogram!!!

12. Definition: Absolute Value of a Complex Number The absolute value, or modulus, of the complex number, where a and b are real numbers, is Imaginary Axis Real Axis

13. A Few More New Formulas The distance between the points u and v in the complex plane: The midpoint of the line segment connecting u and v in the complex plane:

14. Find the distance between u = –4 + i and v = 2 + 5i in the complex plane, and find the midpoint of the segment connecting u and v. Distance: Midpoint: Can we verify these answers graphically?

15. Whiteboard Problems… Write the given complex number in standard form.