Download - Sum and Product Roots
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 .
02 cbxax 0a
1s 2s a
bss
21
a
css 21 *
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