Quadratic Relationships

2.2.1 Quadratic Equations with Complex Solutions

2.2.2 Quadratic Expressions with Complex Factors

2.2.3 Quadratic Structure Equations

Learning Objectives

• Use complex numbers in polynomial identities and equations.

  • Solve quadratic equations with real coefficients that have complex solutions. (2.2.1)

  • (+) Extend polynomial identities to the complex numbers. (2.2.2)

    • For example, rewrite x^2 + 4 as (x + 2i)(x - 2i).

  • (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (2.2.1)

• Interpret the structure of expressions.

  • Use the structure of an expression to identify ways to rewrite it. (2.2.3)

• Solve equations and inequalities in one variable.

  • Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. (2.2.1)

• Solve systems of equations

  • Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x^2 + y^2 = 3. (2.2.1)