Math 12V Notes – Sum and Product of Roots

 

The Sum and Product of Roots of a Quadratic Equation

Given a quadratic equation in general form  then the sum and product of roots are:

               and                  

 

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Sample Question:  Determine the value(s) of k such that  has two roots with a difference of 4.

 

Method #1:  Use the Quadratic Formula

 

From the quadratic equation we know the roots are equal to

 
Since we want the difference to be 4 we have .  Plug values in for a = 4, b = -8k, and c = 9 to get

 

Next, multiply everything by 8 to cancel the denominators to get
 

 

So now simplifying, we have

 

 

Method #2: Use the Sum and Product of Roots

 

Let the roots be p and p + 4 (that makes their difference 4).  Then the sum of the roots is 2p + 4 and the product of the roots is p² + 4p.
In a general quadratic, 

and in our equation,                   

Equate the two sets of info:


Solve the 2nd equation for p (multiply through by 4 and move all terms to one side)....


Plug these values into first equation to find k:
            and      

Which method did you like better?