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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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Completing the Square

Any quadratic equation can be solved using a technique called completing the square.

To use this method, we will rewrite the quadratic equation so that one side is a perfect square. Then we solve using the square root property.

Before we solve a quadratic equation in this way, let’s learn how to complete the square.

To “complete the square” means to transform a binomial of the form x2 + bx into a perfect square trinomial by adding a constant term.

For example, let’s complete the square for x2 + 6x.

We will use rectangles, called algebra tiles, to visualize the process.

• A square tile measuring x units on a side has area x2. We will use this tile to represent x2, the first term of x2 + 6x.

• A rectangular tile that is x units tall and 1 unit wide has an area of 1x. Since 6 · 1x is 6x, we will use six of these tiles to represent the second term of x2 + 6x.

Placed side by side, the tiles form a rectangle that represents x2 + 6x.

Now, we try to rearrange the tiles to form a square. To do this, we move one-half of the tall thin tiles. However, the result is not a complete square because the lower right portion is missing.

The missing piece is a 3-by-3 square.

Thus, to “complete the square,” we must add 3 · 3 = 9 unit tiles.

With the 9 new tiles, the area of the entire square is (x + 3)(x + 3) = (x + 3)2 = x2 + 6x + 9.

By adding 9 to x2 + 6x, we have created the perfect square (x + 3)2.

 

Let’s review the process we used to complete the square.

We moved one-half of the x-tiles.

 x2 + 6x + ?

Multiply the coefficient of x by

 

Then we filled in the remaining space with a square of unit tiles.  Square the result.

32 = 9

We added nine tiles to complete the square.   x2 + 6x + 9

= (x + 3)(x + 3)

= (x + 3)2

This process holds in general:

To find the number needed to complete the square, multiply the coefficient of the x-term by , and then square the result.

 

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