Discriminants

Date: 10/22/96 at 13:0:48
From: Holly Barrow
Subject: Question

Dear Dr. Math,
     
I am a high school student and I have a question for you:

Find the discriminant and determine the nature and the number of the
roots of 2x^2+x-6 = 0 without solving the equation.
     
Thanks for your help in this matter.

Date: 10/22/96 at 17:31:18
From: Doctor Jerry
Subject: Re: Question

Dear Holly Barrow,

Since you mentioned the discriminant, I think you must know that a 
quadratic equation ax^2+bx+c = 0 has roots given by the formula:

  x = (-b + sqrt(b^2-4ac))/(2a) 

  x = (-b - sqrt(b^2-4ac))/(2a)

The expression b^2-4ac underneath the square root sign is the 
discriminant.

  I. If b^2-4ac is positive, then the two roots are real and unequal. 
     For example, the discriminant of the quadratic x^2-5x+6 = 0 is 
     b^2-4ac = 25-4*1*6 = 1.  The roots of this quadratic are 2 and 3.

 II. If b^2-4ac is negative, then the two roots are complex and 
     unequal. For example, the discriminant of the quadratic 
     x^2+x+1 = 0 is b^2-4ac = 1-4*1*1 = -3.  The roots of this 
     quadratic are -1/2 + i*sqrt(3)/2 and -1/2 - i*sqrt(3)/2.

III. If b^2-4ac is zero, then the two roots are real and equal. For 
     example, the discriminant of the quadratic x^2-2x+1 = 0 is 
     b^2-4ac = 4-4*1*1 = 0.  The roots of this quadratic are 1 and 1.

Now, all you need to do is apply this information to your quadratic.  
As a hint, 7^2 is involved.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/22/96 at 18:13:40
From: Doctor Wilkinson
Subject: Re: Question

I don't know what exactly it is that is giving you trouble here, but I 
will try to cover all the possibilites I can think of.  There are 
three aspects to the problem:

(1) What is the discriminant?

(2) What is the relation between the value of the discriminant and 
    the number and nature of the roots?

(3) How do you apply (1) and (2) to the particular example given?

Here are the answers:

(1) The discriminant of a quadratic polynomial ax^2 + bx + c is
    b^2 - 4ac. It is called the discriminant because it discriminates 
    among the various possibilities for the roots of the quadratic 
    equation ax^2 + bx + c = 0.

(2) First of all, what is meant by the nature and number of the roots 
    of a quadratic equation?  

I don't know whether you have had complex numbers yet or not.  It 
makes a difference. Without complex numbers, there are quadratic 
equations with no roots. For example, x^2 + 1 = 0 has no roots, since 
the square of any real number is at least 0, and cannot be -1. But if 
you are allowed to use complex numbers, then i and -i are both roots 
of this equation. It is possible for there to be only one root to a 
quadratic equation. For example, x^2 + 2x + 1 has the root -1 and no 
others, since x^2 + 2x + 1 = (x + 1)^2, and the only way for (x + 1)^2 
to be zero is for x + 1 to be zero. Finally, a quadratic equation can 
have two real roots: for example, x^2 - 3x + 2 has the roots 1 and 2.

So there are three possiblities for the nature and number of roots:

(a) No real roots, in which case there are two complex roots
(b) Just one real root
(c) Two real roots.

Now the neat thing about the discriminant is that it will tell you 
which of these three possibilities is true without having to actually 
find the roots.

There are three possibilites for the discriminant:

(a) If it is < 0, then the equation has two complex roots
(b) If it is = 0, then the equation has one real root
(c) If it is > 0, then the equation has two real roots

Examples:

(a) x^2+1 = 0; a = 1, b = 0, c = 1; b^2 - 4ac = -4, which is < 0, and 
    in fact there are no real roots but two complex ones.

(b) x^2 + 2x + 1 = 0; a -1, b = 2, c = 1; b^2 - 4ac = 0, and there is 
    just one root.

(c) x^2 - 3x +2 - 0; a = 1, b = -3, c = 2; b^2 -4ac = 1, and there are 
    two real roots.

(3) Now let's look at your example.  The equation is

    2x^2 + 2x - 6 = 0.

    Here a = 2, b = 2, c = -6

    b^2 - 4ac = 52

This means that there are two real roots.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/