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TUTORIALS:

Absolute Values
Solving Two-Step Equations Algebraically
Multiplying Monomials
Factoring Trinomials
Solving Quadratic Equations
Power Functions and Transformations
Composition of Functions
Rational Inequalities
Equations of Lines
Graphing Logarithmic Functions
Elimination Using Multiplication
Multiplying Large Numbers
Multiplying by 11
Graphing Absolute Value Inequalities
Polynomials
The Discriminant
Reducing Numerical Fractions to Simplest Form
Addition of Algebraic Fractions
Graphing Inequalities in Two Variables
Adding and Subtracting Rational Expressions with Unlike Denominators
Multiplying Binomials
Graphing Linear Inequalities
Properties of Numbers and Definitions
Factoring Trinomials
Relatively Prime Numbers
Point
Inequalities
Rotating a Hyperbola
Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
Solving Absolute Value Equations
Writing Linear Equations in Slope-Intercept Form
Adding and Subtracting Rational Expressions with Different Denominators
Reducing Rational Expressions
Solving Absolute Value Equations
Equations of a Line - Slope-intercept form
Adding and Subtracting Rational Expressions with Unlike Denominators
Solving Equations with a Fractional Exponent
Simple Trinomials as Products of Binomials
Equivalent Fractions
Multiplying Polynomials
Slope
Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
Solving Cubic Equations by Factoring
Adding and Subtracting Fractions
Multiplying Whole Numbers
Straight Lines
Solving Absolute Value Equations
Solving Nonlinear Equations
Factoring Polynomials by Finding the Greatest Common Factor
Logarithms
Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
Solving Equations
Factoring Polynomials
Solving Quadratic Equations
Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
Square Roots
Adding Polynomials
Arithmetic with Positive and Negative Numbers
Solving Equations
Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
Plotting a Point
Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
Solving Equations with Radicals and Exponents
Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Percents
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Radicals
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
Rationalizing the Denominator
Multiplication by 429
Writing Linear Equations in Point-Slope Form
Multiplying Radicals
Dividing Polynomials by Monomials
Factoring Trinomials
Introduction to Fractions
Square Roots
   
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Evaluating the Quadratic Formula

Solving a Quadratic Equation Graphically

From a graphical point of view, the solutions of a quadratic equation like: 

3· x 2 + 6 · x +1= 0

are the x-coordinates of the points where the graph o f  y = 3· x 2 + 6 · x +1 cuts through the x-axis

(see Figure 1).

Figure 1: The solutions of a quadratic are the xcoordinates of the points where the graph cuts the x-axis.

You can locate these points (at least approximately) by using a graphing calculator.

 

Solving a Quadratic Equation Algebraically

The key difficulty in using a graphing calculator to solve a quadratic equation (in fact, the main problem in using a graphing calculator to solve any equation) is that you need to set the viewing window so that you can see the points where the quadratic graph cuts through the x-axis. Unless you happen to know about where these points will be, setting the viewing window to show them is a hitand- miss, trial-and-error procedure.

As with many of the other equations that we have studied, it is possible to solve a quadratic equation using algebra. Using algebra to solve a quadratic equation avoids the problem of guessing a good size for the viewing window and is often easier because of this. 

 

Using Factoring to Solve a Quadratic Equation

Unless the numbers are chosen to be very nice, factoring a quadratic equation can be very hard or impossible. However, when you are able to factor a quadratic, the factored form makes it very easy for you to find the solutions of the quadratic equation.

Example

Find all solutions of the quadratic equation: 

x 2 + 8 · x +10 = 30.

Solution

First note that solving this quadratic equation is the same as solving the quadratic equation: 

x 2 + 8 · x +10 - 30 = 30 - 30 (Subtract 30 from each side) 

x 2 + 8 · x - 20 = 0 (Simplify)

Solving the quadratic equation  x2 + 8 · x - 20 = 0 will give exactly the same values for x that solving the original quadratic equation,  x2 + 8 · x +10 = 30, will give.

To factor the quadratic, you have to find a pair of numbers that add to give +8 and which multiply to give -20. Two numbers that do this are +10 and -2. So you can write the quadratic equation in factored form as: 

(x - 2) · (x +10) = 0

The two x-values that will satisfy this equation are x = 2 and x = -10. The solutions of the quadratic equation:

x2 + 8 · x +10 = 30

are x = 2 and x = -10.

 

Using Vertex Form to Solve a Quadratic Equation

If you happen to have a quadratic equation where the quadratic part of the equation is already expressed in vertex form, then you can solve the quadratic equation in a reasonably straightforward manner.

If the quadratic part of your quadratic equation is not already expressed in vertex form, then you can convert it to vertex form by completing the square. Completing the square can be quite a complicated process, however, so solving a quadratic equation that is not already in vertex form can be quite an undertaking.

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