Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

MAT-12.AR.11 Solve quadratic equations with real coefficients that have solutions of the form a + bi and a - bi.

MAT-12.NO.06 Know there is a complex number i such that i² = -1, and every complex number has the form of a + bi with a and b real. Understand the hierarchal relationships among subsets of the complex number system.

MAT-12.NO.07 Use the definition i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

MAT-12.NO.08 Use conjugates to find quotients of complex numbers.

MAT-12.NO.09 Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions. Find all solutions to a polynomial equation.

MAT-12.AR.11Solve quadratic equations with real coefficients that have solutions of the form a+bi and a-bi.

MAT-12.NO.10 Represent complex numbers on the complex plane in rectangular, trigonometric, and polar forms. Find the modulus (absolute value) of a complex number. Explain why the rectangular, trigonometric, and polar forms of a given complex number represent the same number.

MAT-12.NO.11 Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.

MAT-12.NO.12 Extend polynomial identities to the complex numbers.

MAT-12.NO.13 Apply the Fundamental Theorem of Algebra to find all roots of a polynomial equation and determine the nature (i.e., integer, rational, irrational, real, complex) of the roots.

Quadratic Equations

MAT-09.AR.10 Solve quadratic equations in one variable by inspection (e.g., for x2 = 49) taking square roots, the quadratic formula, and factoring, as appropriate to the initial form of the equation.

MAT-12.NO.09 Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions. Find all solutions to a polynomial equation.

MAT-12.AR.07 Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.

MAT-12.AR.10 Derive the quadratic formula from the form 0 = ax2 + bx + c.

MAT-12.AR.11Solve quadratic equations with real coefficients that have solutions of the form a + bi and a - bi.

MAT-12.AR.14 Identify zeros of polynomials when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.

MAT-12.AR.17 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.