Algebra Tutorials!

 Tuesday 23rd of January

 Home Solving Linear Equations Systems of Linear Equations Solving Linear Equations Graphically Algebra Expressions Evaluating Expressions and Solving Equations Fraction rules Factoring Quadratic Trinomials Multiplying and Dividing Fractions Dividing Decimals by Whole Numbers Adding and Subtracting Radicals Subtracting Fractions Factoring Polynomials by Grouping Slopes of Perpendicular Lines Linear Equations Roots - Radicals 1 Graph of a Line Sum of the Roots of a Quadratic Writing Linear Equations Using Slope and Point Factoring Trinomials with Leading Coefficient 1 Writing Linear Equations Using Slope and Point Simplifying Expressions with Negative Exponents Solving Equations 3 Solving Quadratic Equations Parent and Family Graphs Collecting Like Terms nth Roots Power of a Quotient Property of Exponents Adding and Subtracting Fractions Percents Solving Linear Systems of Equations by Elimination The Quadratic Formula Fractions and Mixed Numbers Solving Rational Equations Multiplying Special Binomials Rounding Numbers Factoring by Grouping Polar Form of a Complex Number Solving Quadratic Equations Simplifying Complex Fractions Algebra Common Logs Operations on Signed Numbers Multiplying Fractions in General Dividing Polynomials Polynomials Higher Degrees and Variable Exponents Solving Quadratic Inequalities with a Sign Graph Writing a Rational Expression in Lowest Terms Solving Quadratic Inequalities with a Sign Graph Solving Linear Equations The Square of a Binomial Properties of Negative Exponents Inverse Functions fractions Rotating an Ellipse Multiplying Numbers Linear Equations Solving Equations with One Log Term Combining Operations The Ellipse Straight Lines Graphing Inequalities in Two Variables Solving Trigonometric Equations Adding and Subtracting Fractions Simple Trinomials as Products of Binomials Ratios and Proportions Solving Equations Multiplying and Dividing Fractions 2 Rational Numbers Difference of Two Squares Factoring Polynomials by Grouping Solving Equations That Contain Rational Expressions Solving Quadratic Equations Dividing and Subtracting Rational Expressions Square Roots and Real Numbers Order of Operations Solving Nonlinear Equations by Substitution The Distance and Midpoint Formulas Linear Equations Graphing Using x- and y- Intercepts Properties of Exponents Solving Quadratic Equations Solving One-Step Equations Using Algebra Relatively Prime Numbers Solving a Quadratic Inequality with Two Solutions Quadratics Operations on Radicals Factoring a Difference of Two Squares Straight Lines Solving Quadratic Equations by Factoring Graphing Logarithmic Functions Simplifying Expressions Involving Variables Adding Integers Decimals Factoring Completely General Quadratic Trinomials Using Patterns to Multiply Two Binomials Adding and Subtracting Rational Expressions With Unlike Denominators Rational Exponents Horizontal and Vertical Lines
Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Solving Quadratic Equations

## Completing the Square

The Vertex Form of a Quadratic Function

The format for a quadratic equation given above,

y = a Â· x 2 + b Â· x + c, where the letter x represents the input, the letter y represents the value of the output and the letters a, b and c are all numbers, is called standard form.

Other ways of writing the equations for quadratic functions include vertex form,

y = a  Â· (x - h) 2 + k,

where the letter x represents the value of the input, the letter y represents the value of the output and the letters a, h and k all represent numbers. Just as in standard form, in vertex form the number a cannot be equal to zero. Converting a quadratic equation to vertex form is often quite helpful as it allows you to determine exactly where the graph of the quadratic equation reaches its â€œlow pointâ€ or â€œhigh pointâ€ very easily. Every single quadratic formula can be converted to vertex form. The process for doing this conversion is called completing the square.

What the Vertex Form of a Quadratic can tell you about the graph

The vertex form of a quadratic function:

y = a  Â· (x - h) 2 + k,

also tells you whether the graph of the quadratic is smiling or frowning. To check, simply look at the value of a, as you would if the equation had been written in standard form. If the value of a is positive then the quadratic is smiling and if the value of a is negative then the quadratic will be frowning.

The vertex form of a quadratic equation can also tell you about the location of the highest point (on a frowning quadratic) or the lowest point (on a smiling quadratic â€“ see Figure 1 on the next page). This point (the highest point on a frowning quadratic or the lowest point on a smiling quadratic) is called the vertex.

The x-coordinate of the vertex is the number h that appears inside the parentheses of the vertex form and the y-coordinate of the vertex is the number k that appears outside the parentheses in the vertex form.

Figure 1: (a) In this quadratic, a = -1 and the shape of the graph is a â€œfrown.â€ The vertex in this case is the highest point on the graph. (b) In this quadratic a = 0.5 and the shape of the graph is a â€œsmile.â€ The vertex in this case is the lowest point on the graph.

 Copyrights © 2005-2018