Algebra Tutorials!  
     
     
Monday 23rd of October
   
Home
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
Mixed
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Algebra
Order of Operations
Dividing Complex Numbers
Polynomials
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Factoring
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Numbers
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Fractions
Polynomials
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Exponents
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Radicals
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
   
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Solving Quadratic Equations Graphically and Algebraically

A quadratic equation is an equation that can be simplified to follow the pattern: 

y = a · x2 + b· 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. Sometimes the numbers a, b and c are referred to as coefficients.

Solving a Quadratic Equation

Just as you did for linear and power equations, you solve a quadratic equation when you have been given a y-value and need to find all of the corresponding x-values. For example, if you had been given the quadratic equation: 

y = x2 + 8 · x +10,

and the y-value,

y = 30,

then solving the quadratic equation would mean finding all of the numerical values of x that work when you plug them into the equation: 

x2 + 8 · x +10 = 30.

Note that solving this quadratic equation is the same as solving the quadratic equation: 

x2 + 8 · x +10 - 30 = 30 - 30 (Subtract 30 from each side) 

x2 + 8 · x - 20 = 0 (Simplify)

Solving the quadratic equation  x2 + 8 · x - 20 = 0 will give exactly the same values for x that solving the original quadratic equation,  x2 + 8 · x +10 = 30, will give.

The advantage of manipulating the quadratic equation to reduce one side of the equation to zero before attempting to find any values of x is that this manipulation creates a new quadratic equation that can be solved using some fairly standard techniques and formulas.

Solving a Quadratic Function Graphically Using a Graphing Calculator

When you are trying to solve a quadratic equation of the form: 

a · x2 + b · x + c = 0,

the solutions are the x-coordinates of the points where the graph of the quadratic equation  y = a· x2 + b· x + c cuts the horizontal axis.

From a graphical point of view, the solutions of the manipulated quadratic formula: 

x2 + 8 · x - 20 = 0,

are the x-values where the graph of  y = x2 + 8 · x - 20 cuts the horizontal x-axis. (These points, x = 2 and x = -10, are shown in Figure 1).

Figure 1: Graphical representation of the solutions of a quadratic equation.

There may be zero, one or two places where the graph of the quadratic equation  y = a · x2 + b· x + c cuts the horizontal x-axis (see Figure 2, below). This means that the quadratic equation: 

a · x2 + b · x + c = 0,

can have zero, one or two solutions.

You can determine the number of solutions that a quadratic equations has by calculating the discriminant,Δ. The discriminant is equal to: 

Δ = b2 - 4 · a · c .

The sign of the discriminant tells you the number of solutions that exist for the quadratic equation,  a · x2 + b · x + c = 0.

Figure 2: (a) A quadratic equation with zero solutions. (b) A quadratic equation with exactly one solution. (c) A quadratic equation with exactly two solutions.

Sign of discriminant,  Δ = b2 - 4 · a · c Number of solutions of quadratic equation,  a · x2 + b · x + c = 0
Negative (-) Zero solutions
Zero (0) Exactly one solution
Positive (+) Exactly two solutions.

Table 1: Determining the number of solutions using the sign of the discriminant.

 
Copyrights © 2005-2017