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Finding the Greatest Common Factor
Factoring Trinomials
Absolute Value Function
A Summary of Factoring Polynomials
Solving Equations with One Radical Term
Adding Fractions
Subtracting Fractions
The FOIL Method
Graphing Compound Inequalities
Solving Absolute Value Inequalities
Adding and Subtracting Polynomials
Using Slope
Solving Quadratic Equations
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
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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Solving Quadratic Equations

Converting Standard Form to Vertex Form

Completing the Square

You can covert any quadratic formula from standard form:

y = a · x 2 + b · x + c,

to vertex form:

y = a · (x - h) 2 + k,

through an algebraic process called completing the square. The next example demonstrates the steps that are involved in this process.



Convert the quadratic function:

y = 4 · x 2 + 6 · x + 7,

from standard to vertex form and locate the x- and y-coordinates of the vertex.


Once the formula for the quadratic function has been converted to vertex form:

y = a · (x - h) 2 + k,

we can find the vertex by checking the vertex form to find the values of h (which will be the x-coordinate of the vertex) and k (which will be the y-coordinate of the vertex).

Conversion of the formula from standard to vertex form is a four-step process called completing the square.

1. Factor out the coefficient of x2 from all terms.

2. Add and subtract just the right amount1 to create a perfect square

1 To find just the right amount, you take the number that is left multiplying the x after Step 1 has been completed. Whatever this number is, divide the number by 2 and then take the square of what you are left with. This is just the right amount to create a perfect square.

3. Factor the perfect square and combine the constants

4. Distribute the factor that is out in front of the equation

The vertex form of the quadratic function is:

This is not quite the same as the “classic” format of the vertex form:

y = a · (x - h) 2 + k,

because the number inside the parentheses is added to x, rather than subtracted from x. To fix this we can use the fact that the negative of a negative is a positive, so that . Using this to re-write the vertex form in the “classic” format gives:

With the vertex form in the “classic” format you can go ahead and determine the x- and y-coordinates of the vertex. The x-coordinate is and the y-coordinate is

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