Math  201819
AII.3b  Quadratic Equations
AII.3 The student will solve
b) quadratic equations over the set of complex numbers;
BIG IDEAS
 I can find how far and how
fast a bus travels, the number and combinations of fruits that can be
purchased, and how long it takes to drain a swimming pool.
 I will be able to write symbolic representations of the
way numbers behave and will know that in order to maintain equality, an
operation performed on one side must also be performed on the other side.
UNDERSTANDING THE STANDARD
· A quadratic function whose graph does not intersect the xaxis has roots with imaginary components.
· The quadratic formula can be used to solve any quadratic equation.
· The quadratic formula can be derived by applying the completion of squares to any quadratic equation in standard form.
· The value of the discriminant of a quadratic equation can be used to describe the number and type of solutions.
· Solutions of quadratic equations are real or a sum or difference of a real and imaginary component.
· Complex solutions occur in conjugate pairs.
· Quadratic equations with exactly one real root can be referred to as having one distinct root with a multiplicity of two. For instance, the quadratic equation, , has two identical factors, giving one real root with a multiplicity of two.
·
The
definition of absolute value (for any real numbers a and b, where
b 0, if a= b,
then a = b or a = − b) is used in solving absolute value
equations and inequalities.
· Absolute value inequalities in one variable can be solved algebraically using a compound statement.
· Compound statements representing solutions of an inequality in one variable can be represented graphically on a number line.
· Practical problems can be interpreted, represented, and solved using equations and inequalities.
· The process of solving equations can lead to extraneous solutions.
· An extraneous solution is a solution of the simplified form of an equation that does not satisfy the original equation.
· Equations can be solved in a variety of ways.
· The zeros, roots, or solutions of a function are the values of x that make f(x) = 0
· The real zeros of a function are the xintercepts of that function.
· Radical expressions may be converted to expressions using rational exponents.
· The equation of an inverse variation is a rational function.
· Solutions and intervals may be expressed in different formats, including set notation, using equations and inequalities, or interval notation.
 Examples may include:
Equation/ Inequality 
Set Notation 
Interval Notation 
x = 3 
{3} 

x = 3 or x = 5 
{3, 5} 

0 £ x £ 3 
{x0 £ x £ 3} 
[0, 3) 
y ≥ 3 
{y: y ≥ 3} 
[3, ¥) 
Empty (null) set ∅ 
{ } 
ESSENTIALS
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
· AII.3b_{1} Solve a quadratic equation over the set of complex numbers algebraically.
·
AII.3b_{2} Calculate
the discriminant of a quadratic equation to determine the number and type of solutions.
·
AII.3_{1} Solve
equations and verify algebraic
solutions using a graphing utility.
·
AII.3b_{1} Solve
a quadratic equation over the set of complex numbers algebraically.
·
AII.3_{1} Solve
equations and verify algebraic
solutions using a graphing utility.
·
AII.3b_{1} Solve
a quadratic equation over the set of complex numbers algebraically.
· AII.3_{1} Solve
equations and verify algebraic
solutions using a graphing utility.
KEY VOCABULARY
absolute
value equations, absolute value inequalities, quadratic equation, complex
numbers, rational algebraic expression, radical expression, algebraically,
graphically, quadratic function, discriminant, real solutions, complex
solutions, monomial, binomial, denominator, quadratic formula, verify,
completing the square