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 [latex]x = 4[/latex] and [latex]x = 6[/latex].
This means that [latex]x = 4[/latex] (or [latex]x - 4 = 0[/latex]) and [latex]x = 6[/latex] (or [latex]x - 6 = 0[/latex]).
The quadratic equation these roots come from would have as its factored form:
\[(x – 4)(x – 6) = 0\]
All that needs to be done is to multiply these two terms together:
\[(x – 4)(x – 6) = x^2 – 10x + 24 = 0\]
This means that the original equation will be equivalent to [latex]x^2 - 10x + 24 = 0[/latex].
This strategy works for even more complicated equations, such as:
Example 10.8.2
Construct a polynomial equation whose roots are [latex]x = \pm 2[/latex] and [latex]x = 5[/latex].
This means that [latex]x = 2[/latex] (or [latex]x - 2 = 0[/latex]), [latex]x = -2[/latex] (or [latex]x + 2 = 0[/latex]) and [latex]x = 5[/latex] (or [latex]x - 5 = 0[/latex]).
These solutions come from the factored polynomial that looks like:
\[(x – 2)(x + 2)(x – 5) = 0\]
Multiplying these terms together yields:
\[\begin{array}{rrrrcrrrr}
&&(x^2&-&4)(x&-&5)&=&0 \\
x^3&-&5x^2&-&4x&+&20&=&0
\end{array}\]
The original equation will be equivalent to [latex]x^3 - 5x^2 - 4x + 20 = 0[/latex].
Caveat: the exact form of the original equation cannot be recreated; only the equivalent. For example, [latex]x^3 - 5x^2 - 4x + 20 = 0[/latex] is the same as [latex]2x^3 - 10x^2 - 8x + 40 = 0[/latex], [latex]3x^3 - 15x^2 - 12x + 60 = 0[/latex], [latex]4x^3 - 20x^2 - 16x + 80 = 0[/latex], [latex]5x^3 - 25x^2 - 20x + 100 = 0[/latex], 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).
- 2, 5
- 3, 6
- 20, 2
- 13, 1
- 4, 4
- 0, 9
- [latex]\dfrac{3}{4}, \dfrac{1}{4}[/latex]
- [latex]\dfrac{5}{8}, \dfrac{5}{7}[/latex]
- [latex]\dfrac{1}{2}, \dfrac{1}{3}[/latex]
- [latex]\dfrac{1}{2}, \dfrac{2}{3}[/latex]
- ± 5
- ± 1
- [latex]\pm \dfrac{1}{5}[/latex]
- [latex]\pm \sqrt{7}[/latex]
- [latex]\pm \sqrt{11}[/latex]
- [latex]\pm 2\sqrt{3}[/latex]
- 3, 5, 8
- −4, 0, 4
- −9, −6, −2
- ± 1, 5
- ± 2, ± 5
- [latex]\pm 2\sqrt{3}, \pm \sqrt{5}[/latex]
