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 x = 4 and x = 6.

This means that x = 4 (or x - 4 = 0) and x = 6 (or x - 6 = 0).

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 x^2 - 10x + 24 = 0.

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

Example 10.8.2

Construct a polynomial equation whose roots are x = \pm 2 and x = 5.

This means that x = 2 (or x - 2 = 0), x = -2 (or x + 2 = 0) and x = 5 (or x - 5 = 0).

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 x^3 - 5x^2 - 4x + 20 = 0.

Caveat:  the exact form of the original equation cannot be recreated; only the equivalent. For example, x^3 - 5x^2 - 4x + 20 = 0 is the same as 2x^3 - 10x^2 - 8x + 40 = 0, 3x^3 - 15x^2 - 12x + 60 = 0, 4x^3 - 20x^2 - 16x + 80 = 0, 5x^3 - 25x^2 - 20x + 100 = 0, 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. \dfrac{3}{4}, \dfrac{1}{4}
  8. \dfrac{5}{8}, \dfrac{5}{7}
  9. \dfrac{1}{2}, \dfrac{1}{3}
  10. \dfrac{1}{2}, \dfrac{2}{3}
  11. ± 5
  12. ± 1
  13. \pm \dfrac{1}{5}
  14. \pm \sqrt{7}
  15. \pm \sqrt{11}
  16. \pm 2\sqrt{3}
  17. 3, 5, 8
  18. −4, 0, 4
  19. −9, −6, −2
  20. ± 1, 5
  21. ± 2, ± 5
  22. \pm 2\sqrt{3}, \pm \sqrt{5}

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-10-8/”>Answer Key 10.8

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