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	Arithmetic Operations with Numerical Fractions
	Multiplying a Polynomial by a Monomial
	Solving Linear Equation
	Solving Linear Equations
	Solving Inequalities
	Solving Compound Inequalities
	Solving Systems of Equations Using Substitution
	Simplifying Fractions 3
	Factoring quadratics
	Special Products
	Writing Fractions as Percents
	Using Patterns to Multiply Two Binomials
	Adding and Subtracting Fractions
	Solving Linear Inequalities
	Adding Fractions
	Solving Systems of Equations -
	Exponential Functions
	Integer Exponents
	Example 6
	Dividing Monomials
	Multiplication can Increase or Decrease a Number
	Graphing Horizontal Lines
	Simplification of Expressions Containing only Monomials
	Decimal Numbers
	Negative Numbers
	Factoring
	Subtracting Polynomials
	Adding and Subtracting Fractions
	Powers of i
	Multiplying and Dividing Fractions
	Simplifying Complex Fractions
	Finding the Coordinates of a Point
	Fractions and Decimals
	Rational Expressions
	Solving Equations by Factoring
	Slope of a Line
	Percent Introduced
	Reducing Rational Expressions to Lowest Terms
	The Hyperbola
	Standard Form for the Equation of a Line
	Multiplication by 75
	Solving Quadratic Equations Using the Quadratic Formula
	Raising a Product to a Power
	Solving Equations with Log Terms on Each Side
	Monomial Factors
	Solving Inequalities with Fractions and Parentheses
	Division Property of Square and Cube Roots
	Multiplying Two Numbers Close to but less than 100
	Solving Absolute Value Inequalities
	Equations of Circles
	Percents and Decimals
	Integral Exponents
	Linear Equations - Positive and Negative Slopes
	Multiplying Radicals
	Factoring Special Quadratic Polynomials
	Simplifying Rational Expressions
	Adding and Subtracting Unlike Fractions
	Graphuing Linear Inequalities
	Linear Functions
	Solving Quadratic Equations by Using the Quadratic Formula
	Adding and Subtracting Polynomials
	Adding and Subtracting Functions
	Basic Algebraic Operations and Simplification
	Simplifying Complex Fractions
	Axis of Symmetry and Vertices
	Factoring Polynomials with Four Terms
	Evaluation of Simple Formulas
	Graphing Systems of Equations
	Scientific Notation
	Lines and Equations
	Horizontal and Vertical Lines
	Solving Equations by Factoring
	Solving Systems of Linear Inequalities
	Adding and Subtracting Rational Expressions with Different Denominators
	Adding and Subtracting Fractions
	Solving Linear Equations
	Simple Trinomials as Products of Binomials
	Solving Nonlinear Equations by Factoring
	Solving System of Equations
	Exponential Functions
	Computing the Area of Circles
	The Standard Form of a Quadratic Equation
	The Discriminant
	Dividing Monomials Using the Quotient Rule
	Squaring a Difference
	Changing the Sign of an Exponent
	Adding Fractions
	Powers of Radical Expressions
	Steps for Solving Linear Equations
	Quadratic Expressions Complete Squares
	Fractions 1
	Properties of Negative Exponents
	Factoring Perfect Square Trinomials
	Algebra
	Solving Quadratic Equations Using the Square Root Property
	Dividing Rational Expressions
	Quadratic Equations with Imaginary Solutions
	Factoring Trinomials Using Patterns

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The Discriminant

Example 1

Use the discriminant to determine the nature of the solutions of this quadratic equation: 5x² - 3x + 8 = 0

Solution

The equation has the form ax² + bx + c = 0 where a = 5, b= -3, and c = 8.

The equation has the form ax² + bx + c = 0 where a = 5, b = -3, and c = 8.

Substitute the values of a, b, and c into the discriminant and simplify.

b² - 4ac

= (-3)² - 4(5)(8)

= 9 - 160

= -151

The discriminant is -151, a negative number.

So the equation 5x² - 3x + 8 = 0 has no real number solutions.

Example 2

Use the discriminant to determine the nature of the solutions of this quadratic equation: 9x² - 6x = -1

Solution

To put the equation in standard form, add 1 to both sides of the equation.

Now the equation has the form ax² + bx + c = 0 where a = 9, b = - 6, and c = 1.

9x² - 6x = -1

9x² - 6x + 1 = 0

Substitute the values of a, b, and c into the discriminant and simplify.

b² - 4ac

= (-6)2 - 4(9)(1)

= 36 - 36

= 0

The discriminant is 0.

So the equation 9x² - 6x = -1 has two identical real number solutions.