# AII.4b and *AII.3b - Quadratic Equations

AII.4    The student will solve, algebraically and graphically,

b)  quadratic equations over the set of complex numbers;

Graphing calculators will be used for solving and for confirming the algebraic solutions.

AII.3  The student will solve

b)  quadratic equations over the set of complex numbers;

Bloom's Level:  Analyze

### 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 x-axis has roots with imaginary components.
• The quadratic formula can be used to solve any quadratic equation.
• The value of the discriminant of a quadratic equation can be used to describe the number of real and complex solutions.
• 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 can be solved graphically or by using a compound statement.
• Real-world problems can be interpreted, represented, and solved using equations and inequalities.
• The process of solving radical or rational equations can lead to extraneous solutions.
• Equations can be solved in a variety of ways.
• Set builder notation may be used to represent solution sets of equations and inequalities.

·  A quadratic function whose graph does not intersect the x-axis 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 x-intercepts 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 {x|0 £ x £ 3} [0, 3) y ≥ 3 {y: y ≥ 3} [3, ¥) Empty (null) set ∅ { }

### ESSENTIALS

Equations

Formula & Discriminant

All.4b1  Solve a quadratic equation over the set of complex numbers using an appropriate strategy.

·  AII.3b1  Solve a quadratic equation over the set of complex numbers algebraically.

All.4b2  Calculate the discriminant of a quadratic equation to determine the number of real and complex solutions.

·  AII.3b2  Calculate the discriminant of a quadratic equation to determine the number and type of solutions.

AII.41  Apply an appropriate equation to solve a real-world problem.

·  AII.31  Solve equations and verify algebraic solutions using a graphing utility.

Factoring & Graphing

All.4b1  Solve a quadratic equation over the set of complex numbers using an appropriate strategy.

·  AII.3b1  Solve a quadratic equation over the set of complex numbers algebraically.

AII.41  Apply an appropriate equation to solve a real-world problem.

·  AII.31  Solve equations and verify algebraic solutions using a graphing utility.

Completing the Square

All.4b1  Solve a quadratic equation over the set of complex numbers using an appropriate strategy.

·  AII.3b1  Solve a quadratic equation over the set of complex numbers algebraically.

AII.41   Apply an appropriate equation to solve a real-world problem.

·  AII.31  Solve equations and verify algebraic solutions using a graphing utility.

All.4b3  Recognize that the quadratic formula can be derived by applying the completion of squares to a quadratic equation in standard form.

### 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

Updated: Oct 27, 2017