Algebra Tutorials!  
Monday 15th of July
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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Solving Quadratic Equations by Graphing

Objective Help you understand that the solutions of a quadratic equation occur where the graph of the corresponding function intersects the x-axis.

In this lesson, you should be able to use the graphing techniques you already learned to help you solve or approximate solutions to quadratic equations. Let's begin by stating a definition.

Quadratic Equations

Quadratic Equation

A quadratic equation is an equation of the form f ( x ) = 0, where f ( x ) = ax 2 + bx + c is a quadratic function.

The goal in solving a quadratic equation is to find what x values make the y value of the quadratic function y = f ( x ) equal to zero. The y value of the function will be zero where the graph intersects the x-axis. Geometrically, this is because a solution of an equation f ( x ) = 0 occurs when the graph of the function y = f ( x ) intersects the line y = 0. But the line y = 0 is the x-axis.

Recall that the graph of a quadratic function is a parabola. So the solutions of a quadratic equation occur where the parabola representing the graph of the quadratic intersects the x-axis.

Try to draw some parabolas like the following, and observe that a parabola can either

• not intersect the x-axis,

• intersect the x -axis in exactly one point, or

• intersect the x -axis in two points.

1. There may be no real solutions. This will occur if the parabola does not intersect the x-axis. Either the parabola opens upwards and the vertex (a minimum) lies above the x -axis, or the parabola opens downwards and the vertex (a maximum) lies below the x-axis.

2. There may be one solution. This occurs when the parabola intersects the x -axis in exactly one point. This happens when the vertex of the parabola lies on the x -axis.

3. There may be two solutions. This occurs when the parabola intersects the x -axis in two points. Either the parabola opens upward and the vertex lies below the x -axis, or the parabola opens downward and the vertex lies above the x -axis.

Keep in mind that a quadratic equation may have no solutions, one solution, or two solutions, because a parabola can intersect the x -axis in zero, one, or two points.

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