HOME
solving equations fractions 2
solving quadratic equations 4
equations 4
solving equations 9
linear equations 3
solving equations 6
solving equations 12
solving equations 8
linear equations 2
solving linear equations 2
solving linear equations 4
solving equations 13
solving quadratic equations 2
solving equations 10
solving equations 7
solving linear equations 3
quadratic equations 2
linear equations 4
solving equations 11
solving rational equations 2
equations 2
solving equations 4
solving systems linear equations 2
solving linear equations 5
solving equations 3
solving quadratic equations 3
solving quadratic equations 5
equations 3
solving equations 5
solving equations 2

Quadratic Equations

Quadratic Equations

Using the Discriminant to predict the roots of a quadratic equation

The discriminant of a quadratic equation is the value under the square root sign in the quadratic formula.

Remember the quadratic formula for an equation in the form ax + bx + c = 0 is:

From this formula the discriminant is: b - 4ac

When you evaluate the discriminant for a quadratic equation, if the result is:

positive You will have 2 different real solutions to the equation
  If this number is a perfect square number, there will be 2 different rational answers. If this number is a not perfect square number, there will be 2 different irrational answers.
zero You will have 1 real, rational solution to the equation - that is, there will be a repeated answer
negative You will have no real solutions to the equation (only imaginary answers)

 

Examples:

Use the discriminant to predict the roots of the following equations:

1. x + 7x + 12 = 0 a = 1 b = 7 c = 12 b - 4ac = 7 - 4(1)(12) = 49 - 48 = 1

Since the result is positive, there should be 2 different real solutions.

In fact, there will be 2 different rational solutions because 1 is a perfect square number.

(Perfect square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc)

2. x + 7x + 3 = 0 a = 1 b = 7 c = 3 b - 4ac = 7 - 4(1)(3) = 49 - 12 = 37

Since the result is positive, there should be 2 different real solutions.

In fact, there will be 2 different irrational solutions because 37 is not a perfect square number.

3. x + 4x + 4 = 0 a = 1 b = 4 c = 4 b - 4ac = 4 - 4(1)(4) = 16 - 16 = 0

Since the result is zero, there should be only one real, rational solution

4. x - x + 4 = 0 a = 1 b = -1 c = 4 b - 4ac = (-1) - 4(1)(4) = 1 - 16 = -15

Since the result is negative, there should be no real solutions.

Buy  Algebrator now: 

Instant download and optional CD

Only $39.99

Click to Buy Now:



OR

Attention: We are currently running a special promotional offer for equation-solver.com Visitors -- if you order Algebrator by midnight of February 8th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35.00 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minutes of live math tutoring from tutor.com!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebrator is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebrator solve it:

Step 3 : Ask for an explanation for the steps you don't understand:

Algebrator can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • operations with complex numbers (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

Buy  Algebrator now: 

Instant download and optional CD

Only $39.99

Click to Buy Now:



OR











































2010-02-08 02:14:15