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The Discriminant
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Addition of Algebraic Fractions
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Properties of Numbers and Definitions
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Relatively Prime Numbers
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Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
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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
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Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
Solving Cubic Equations by Factoring
Adding and Subtracting Fractions
Multiplying Whole Numbers
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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
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Solving Quadratic Equations by Completing the Square

Example

A rectangular patio is 12 feet long and 10 feet wide. We want to increase the length and the width so the area of the new patio will be twice that of the original patio. The length will be increased by twice the amount that the width is increased. By how much should we increase each dimension?

Solution

Calculate the area of the original patio.

 

The area of the new patio will be twice that of the original patio.

 original area

 

 new area

= (12 feet)(10 feet)

= 120 ft2

= 2 · 120 ft2

= 240 ft2

Let x = the number of feet the width should be increased.
To represent the new width, add x to the original width.

Since the length will be increased by twice as much as the width, to represent the new length, add 2x to the original length.

 new width

 

new length

= 10 + x ft

 

= 12 + 2x ft

The area of the rectangle is:

Area = (length)(width)

Now, write an equation for the area of the new patio.

Substitute the expressions.

Multiply the binomials.

Simplify.

 new area 

240

240

240

= (new length)(new width)

= (12 + 2x)(10 + x)

= 120 + 12x + 20x + 2x2

= 120 + 32x + 2x2

This quadratic equation can be solved by completing the square.

Step 1 Isolate the x2-term and the x-term on one side of the equation.

Subtract 120 from both sides.

120 = 32x + 2x2
Write the equation with decreasing powers of x on the left. 2x2 + 32x = 120
Step 2 If the coefficient of x2 is not 1, divide both sides of the equation by the coefficient of x2.

The coefficient of x2 is 2.

Divide both sides of the equation by 2.

x2 + 16x = 60
Step 3 Find the number that completes the square: Multiply the coefficient of x by . Square the result.

The coefficient of the x-term is 16.

Step 4 Add the result of Step 3 to both sides of the equation.

Add 64 to both sides of the equation.

 x2 + 16x + 64 = 60 + 64
Step 5 Write the trinomial as the square of a binomial.

Write x2 + 16x + 64 as the square of a binomial.

Also, simplify the right side of the equation.

(x + 8)2 = 124
Step 6 Finish solving using the Square Root Property.

Use the Square Root Property.

For each equation, subtract 8 from both sides.

Step 7 Check each solution.

We leave the check for you.

 

 

You can simplify the radical:

So the solutions are:

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