Nature of Roots
(Sum and Product)
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Solving quadratic equations by factoring, such as the example at the right, is a well honed skill at this point in your mathematical career.  

But did you ever stop to notice how the roots of equations are related to the coefficients and constants of the equation itself?

       Let's investigate:

Consider the general quadratic equation:
           where

 

Multiply to create a leading coefficient of 1:
              

                   

Represent the roots of the equation as  and  :
               
 

is the sum of the roots

is the product of the roots
Comparing the equations, it can be seen that:
         or     

                    and       

Sum of the roots:

Product of the roots:


Our investigation reveals that there is a definite relationship between the roots of a quadratic equation and the coefficient of the second term and the constant term.

The sum of the roots of a quadratic equation is equal to the negation of the
coefficient of the second term divided by the leading coefficient.

The product of the roots of a quadratic equation is equal to the
constant term divided by the leading coefficient.

You will discover, as you progress in your mathematical career, that these types of
relationships also extend to equations of higher degree.


Example:  Write a quadratic equation whose roots are -3 and .

Of course, this question could be answered by simply multiplying the factors formed by these roots:
     

But with our new found discoveries, we can also arrive at the answer by utilizing the relationship between the roots and coefficients and constants.
 

The sum of the roots is .
   So the coefficient of the second term will be (the negation of the sum).		 The product of the roots is .

So the constant term will be .

Answer:
       The equation is:  

Multiplying by 2 gives: