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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   
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Solving Quadratic Equations by Graphing

Objective Understand why the solutions of a quadratic equation occur where the graph of the corresponding function intersects the x-axis.

Let's see some examples, where the goal is to find or approximate the solutions to a quadratic equation ax 2 + bx + c = 0 by looking to see where the graph of the corresponding quadratic equation intersects the x-axis.

 

Example 1

Solve x 2 - x - 2 = 0 by graphing.

Solution

Step 1 Make a table of values, and then graph the function y = x 2 - x - 2.

Step 2 Find where the parabola intersects the x -axis. In this case, this occurs when x = -1 and x = 2. Thus, - 1 and 2 are the solutions to the equation x 2 - x - 2 = 0 .

Step 3 Check the answer by factoring.

0 = x 2 - x - 2

0 = ( x + 1)( x - 2)

Set each factor equal to zero.

x + 1 = 0 x - 2 = 0
x = -1 x + 2

So, the solutions check.

 

Example 2

Solve x 2 - 2x - 2 = 0 by graphing.

Solution

Step 1 Make a table of values, and then graph the function y = x 2 - 2x - 2.

Step 2 Notice that the parabola does not intersect the x-axis. This parabola opens upward and we can compute that the vertex is at (1, 1). This vertex lies above the x-axis, so the parabola never intersects the x-axis. We can conclude that there are no solutions to the equation x 2 - 2x - 2 = 0.

 

Example 3

Solve by graphing.

Solution

Step 1 Make a table of values, and then graph the function .

Step 2 Try to factor and solve for the roots. In this case, does not have integral factors.

Step 3 Use the graph to approximate the solutions. We can see from the graph that the parabola intersects the x -axis twice: once between -3 and -2, and once roughly halfway between 0 and 1. We can therefore estimate solutions to the equation to be . (In this case, these two estimates are actually exact solutions.)

In general, graphs provide a good way of approximating solutions to quadratic equations when the corresponding quadratic expressions cannot be factored with integral factors. In order to get exact solutions, we need to use the Quadratic Formula, which will be discussed in other lessons.

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