Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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The Quadratic Formula

It is possible to find the solutions of a quadratic equation without having to graph and use a calculator, without having to factor and without having to complete the square. The key to being able to find the solutions of a quadratic equation is a relationship called the quadratic formula.

The values of x that solve the quadratic equation: 

a · x2 + b · x + c = 0,


Because of the similarity in the two formulas, they are often combined (by writing “±” in place of the “+” and “-”) into a single formula that is known as the quadratic formula:


Find all of the solutions of the following quadratic equation:

x2 + 8 · x - 20 = 0.


In this equation, the values of the coefficients a, b and c are:

a = 1

b = 8

c = -20.

Plugging these values into the first version of the quadratic equation (the version with the “+” in it) gives:

Plugging these values into the second version of the quadratic equation (the version with the “-” in it) gives: 

 There are two solutions of the quadratic equation  x2 + 8 · x - 20 = 0, and they are x = 2 and x = -10.


Order of Operations and the Quadratic Formula

The quadratic formula:

is the most complicated formula that we have done in the course so far. This formula involves parentheses, exponents, multiplication, division, addition and subtraction. When you use the quadratic formula to find the solutions of a quadratic equation you have to pay attention to the order of operations to ensure that you evaluate the formula correctly.

This principle of working out the power (or exponent) first, and leaving the multiplication until you have finished the power (or exponent) is part of a system of rules for evaluating complicated equations. The rules are called the Order of Operations.

• B: Work out everything in brackets/parentheses first.

• E: Next, work out all of the exponents.

• M/D: When you have finished with the exponents, do all of the multiplication and division.

• A/S: Finally, do all of the addition and subtraction.

Some people use a mnemonic device to remember the order that the operations:

Parentheses Please
Exponents Excuse
Multiplication My
Division Dear
Addition Aunt
Subtraction Sally.

This applies to the quadratic formula by stipulating that you should evaluate the quadratic formula in the following order:

• First: Work out b2 - 4"a"c first.

• Second: Take the square root of b2 - 4 · a · c.

• Third: Add or subtract this square root from -b.

• Fourth: Divide by 2 · a.

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