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# Complex Solutions to Quadratic Equations

Lesson 4 of 16

## Objective: SWBAT solve quadratic equations with real coefficients that have complex solutions, graph the solutions in the complex plane, and recognize complex conjugates.

## Big Idea: Students investigate the geometry of the complex solutions to a quadratic equation in a dynamic setting.

*45 minutes*

Once the symmetry of the complex conjugate solutions to the quadratic equations has been discussed and understood, I give the class another ten minutes to wrap-up the remaining two problems which exemplify **math practice standards 7 and 8**. To be successful with these problems, most students will need some fairly strong guidance in the form of timely hints or questions to point their thinking in the right direction.

Problem 3 asks students to carefully examine the structure of a factored quadratic and recognize that once the roots of any quadratic are known - complex or real - that quadratic can be written in a factored form. They should see the factored form as (x - root)(x - root). Making use of structure **(MP7)** like this is a powerful habit of thought!

Problem 4 asks students to apply this new notion of factoring to the problem of the sum of two squares **(MP8)**. They should recall from previous courses that the difference of two squares is always factorable into the sum and difference of the roots. Now they will see that the *sum* is always factorable into a pair of complex conjugates. Analogies like these can make mathematic come alive with beauty for many students!

**Math Tip**: To factor a sum of two squares, they might set the sum of two squares equal to zero, find the solutions, and then write the factored form. Alternatively, they might factor out (-1), in the form of *i*^2, before factoring the resulting difference of two squares.

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#### Homework

*3 min*

Regardless of whether or not everyone has completed problems 3 and 4, I call a halt to all work a couple of minutes before the period ends.

Tonight's homework asks students to identify the complex solutions to a quadratic equation and then graph those solutions in the complex number plane. After what the students have done today, this should be fairly straightforward practice. I'll remind them that they're free to use the Quadratic formula, the method of completing the square, or they may find complex factors for the equation.

Additionally, some students may still need to finish up the classwork problems from today's lesson. They should make sure they see me in study hall for extra help if they need it! I'm always free, and I get awfully lonely if no one comes to talk about math!

#### Resources

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: Quadratic Equations, Day 1 of 2
- LESSON 2: Quadratic Equations, Day 2 of 2
- LESSON 3: Inconceivable! The Origins of Imaginary Numbers
- LESSON 4: Complex Solutions to Quadratic Equations
- LESSON 5: Complex Addition
- LESSON 6: The Parallelogram Rule
- LESSON 7: Complex Arithmetic and Vectors
- LESSON 8: Multiplying Complex Numbers, Day 1 of 4
- LESSON 9: Multiplying Complex Numbers, Day 2 of 4
- LESSON 10: Muliplying Complex Numbers, Day 3 of 4
- LESSON 11: Multiplying Complex Numbers, Day 4 of 4
- LESSON 12: Practice & Review
- LESSON 13: Dividing Complex Numbers
- LESSON 14: Quadratic Functions Revisited, Day 1
- LESSON 15: Quadratic Functions Revisited, Day 2
- LESSON 16: Complex Numbers Test