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:

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:

[latex](x - 4)(x - 6) = 0[/latex]

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

[latex](x - 4)(x - 6) = x^2 - 10x + 24 = 0[/latex]

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:

[latex](x - 2)(x + 2)(x - 5) = 0[/latex]

Multiplying these terms together yields:

[latex]\begin{array}{rrrrcrrrr} &&(x^2&-&4)(x&-&5)&=&0 \\ x^3&-&5x^2&-&4x&+&20&=&0 \end{array}[/latex]

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).

  1. 2, 5
  2. 3, 6
  3. 20, 2
  4. 13, 1
  5. 4, 4
  6. 0, 9
  7. [latex]\dfrac{3}{4}, \dfrac{1}{4}[/latex]
  8. [latex]\dfrac{5}{8}, \dfrac{5}{7}[/latex]
  9. [latex]\dfrac{1}{2}, \dfrac{1}{3}[/latex]
  10. [latex]\dfrac{1}{2}, \dfrac{2}{3}[/latex]
  11. ± 5
  12. ± 1
  13. [latex]\pm \dfrac{1}{5}[/latex]
  14. [latex]\pm \sqrt{7}[/latex]
  15. [latex]\pm \sqrt{11}[/latex]
  16. [latex]\pm 2\sqrt{3}[/latex]
  17. 3, 5, 8
  18. −4, 0, 4
  19. −9, −6, −2
  20. ± 1, 5
  21. ± 2, ± 5
  22. [latex]\pm 2\sqrt{3}, \pm \sqrt{5}[/latex]

Answer Key 10.8

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