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Finding the Greatest Common Factor
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Absolute Value Function
A Summary of Factoring Polynomials
Solving Equations with One Radical Term
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The FOIL Method
Graphing Compound Inequalities
Solving Absolute Value Inequalities
Adding and Subtracting Polynomials
Using Slope
Solving Quadratic Equations
Factoring
Multiplication Properties of Exponents
Completing the Square
Solving Systems of Equations by using the Substitution Method
Combining Like Radical Terms
Elimination Using Multiplication
Solving Equations
Pythagoras' Theorem 1
Finding the Least Common Multiples
Multiplying and Dividing in Scientific Notation
Adding and Subtracting Fractions
Solving Quadratic Equations
Adding and Subtracting Fractions
Multiplication by 111
Adding Fractions
Multiplying and Dividing Rational Numbers
Multiplication by 50
Solving Linear Inequalities in One Variable
Simplifying Cube Roots That Contain Integers
Graphing Compound Inequalities
Simple Trinomials as Products of Binomials
Writing Linear Equations in Slope-Intercept Form
Solving Linear Equations
Lines and Equations
The Intercepts of a Parabola
Absolute Value Function
Solving Equations
Solving Compound Linear Inequalities
Complex Numbers
Factoring the Difference of Two Squares
Multiplying and Dividing Rational Expressions
Adding and Subtracting Radicals
Multiplying and Dividing Signed Numbers
Solving Systems of Equations
Factoring Out the Opposite of the GCF
Multiplying Special Polynomials
Properties of Exponents
Scientific Notation
Multiplying Rational Expressions
Adding and Subtracting Rational Expressions With Unlike Denominators
Multiplication by 25
Decimals to Fractions
Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
Slopes
Graphing Lines in the Coordinate Plane
Graphing Functions
Powers of Ten
Zero Power Property of Exponents
The Vertex of a Parabola
Rationalizing the Denominator
Test for Factorability for Quadratic Trinomials
Trinomial Squares
Solving Two-Step Equations
Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
Solving Polynomial Equations by Factoring
Laws of Exponents
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Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
 
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Completing the Square

The essential part of completing the square is to recognize a perfect square trinomial when given its first two terms. For example, if we are given x2 + 6x, how do we recognize that these are the first two terms of the perfect square trinomial x2 + 6x + 9? To answer this question, recall that x2 + 6x + 9 is a perfect square trinomial because it is the square of the binomial x + 3:

(x + 3)2 = x2 + 2 · 3x + 32 = x2 + 6x + 9

Notice that the 6 comes frommultiplying 3 by 2 and the 9 comes from squaring the 3. So to find the missing 9 in x2 + 6x, divide 6 by 2 to get 3, then square 3 to get 9. This procedure can be used to find the last term in any perfect square trinomial in which the coefficient of x2 is 1.

 

Rule for Finding the Last Term

The last term of a perfect square trinomial is the square of one-half of the coefficient of the middle term. In symbols, the perfect square trinomial whose first two terms are

 .

 

Helpful hint

Review the rule for squaring a binomial: square the first term, find twice the product of the two terms, then square the last term. If you are still using FOIL to find the square of a binomial, it is time to learn the proper rule.

 

Example 1

Finding the last term

Find the perfect square trinomial whose first two terms are given.

Solution

a) One-half of 8 is 4, and 4 squared is 16. So the perfect square trinomial is x2 + 8x + 16.

b) One-half of -5 is , and squared is . So the perfect square trinomial is .

c) One-half of is , and squared is . So the perfect square trinomial is

d) One-half of is , and . So the perfect square trinomial is

Another essential step in completing the square is to write the perfect square trinomial as the square of a binomial. Recall that

a2 + 2ab + b2 = (a + b)2 and a2 - 2ab + b2 = (a - b)2.

 

Example 2

Factoring perfect square trinomials

Factor each trinomial.

Solution

a) The trinomial x2 + 12x + 36 is of the form a2 + 2ab + b2 with a = x and b = 6. So x2 + 12x + 36 = (x + 6)2. Check by squaring x + 6.

b) The trinomial is of the form a2 - 2ab + b2 with a = y and . So

Check by squaring .

c) The trinomial is of the form a2 - 2ab + b2 with a = z and . So

 

 

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