# 10.8 Construct a Quadratic Equation from its Roots

It is possible to construct an equation from its roots, and the process is surprisingly simple. Consider the following:

Example 10.8.1

Construct a quadratic equation whose roots are and .

This means that (or ) and (or ).

The quadratic equation these roots come from would have as its factored form:

All that needs to be done is to multiply these two terms together:

This means that the original equation will be equivalent to .

This strategy works for even more complicated equations, such as:

Example 10.8.2

Construct a polynomial equation whose roots are and .

This means that (or ), (or ) and (or ).

These solutions come from the factored polynomial that looks like:

Multiplying these terms together yields:

The original equation will be equivalent to .

Caveat:  the exact form of the original equation cannot be recreated; only the equivalent. For example, is the same as , , , , and so on. There simply is not enough information given to recreate the exact original—only an equation that is equivalent.

# Questions

Construct a quadratic equation from its solution(s).

1. 2, 5
2. 3, 6
3. 20, 2
4. 13, 1
5. 4, 4
6. 0, 9
7. ± 5
8. ± 1
9. 3, 5, 8
10. −4, 0, 4
11. −9, −6, −2
12. ± 1, 5
13. ± 2, ± 5