Sum and Product Roots

Lesson 6-5

The Sum and the Product Roots Theorem

   In a quadratic whose leading coefficient is 1:

• the sum of the roots is the negative of the coefficient of x;

• the product of the roots is the constant term.

Sum and Product of Roots

If the roots of with

are and , then

and .

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Example 1

   Construct the quadratic whose roots are 2 and 3.

 

Solution.   The sum of the roots is 5, their product is 6, therefore the quadratic is  x² − 5x + 6.

The sum of the roots is the negative of the coefficient of x.  The product of the roots is the constant term.

Example 2

   Construct the quadratic whose roots are 2 + ,

 2 − .

Solution.   The sum of the roots is 4.   Their product is the Difference of two squares:

 2² − ( )² = 4 − 3 = 1.

The quadratic therefore is  x² − 4x + 1.

3

3

3

Example 3

  Construct the quadratic whose roots are 2 + 3i,  2 − 3i, where i is the complex unit.   The sum of the roots is 4.   The product

again is the Difference of Two Squares:  4 − 9i² = 4 + 9 = 13.

The quadratic with those roots is  x² − 4x + 13.

Example 4

   Construct the quadratic whose roots are −3, 4.

The sum of the roots is 1.  Their product is −12.  

Therefore, the quadratic is  x² − x − 12.

Example 5

  Construct the quadratic whose roots are  3 + , 3 − .

The sum of the roots is 6.  Their product is 9 − 3 = 6.

Therefore, the quadratic is  x² − 6x + 6.

3

3

Example 6

   Construct the quadratic whose roots are  2 + i ,  2 − i .

The sum of the roots is 4.  Their product is

4 − ( i )² = 4 + 5 = 9.

Therefore, the quadratic is  x² − 4x + 9.

5

5

5