Chapter 10: Quadratics
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:
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:
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.
Construct a quadratic equation from its solution(s).
- 2, 5
- 3, 6
- 20, 2
- 13, 1
- 4, 4
- 0, 9
- ± 5
- ± 1
- 3, 5, 8
- −4, 0, 4
- −9, −6, −2
- ± 1, 5
- ± 2, ± 5
<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-10-8/”>Answer Key 10.8