10.8 Construct a Quadratic Equation from its Roots

Terrance Berg

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:

$(x - 4) (x - 6) = x^{2} - 10 x + 24 = 0$

This means that the original equation will be equivalent to $x^{2} - 10 x + 24 = 0$ .

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:

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

The original equation will be equivalent to $x^{3} - 5 x^{2} - 4 x + 20 = 0$ .

Caveat: the exact form of the original equation cannot be recreated; only the equivalent. For example, $x^{3} - 5 x^{2} - 4 x + 20 = 0$ is the same as $2 x^{3} - 10 x^{2} - 8 x + 40 = 0$ , $3 x^{3} - 15 x^{2} - 12 x + 60 = 0$ , $4 x^{3} - 20 x^{2} - 16 x + 80 = 0$ , $5 x^{3} - 25 x^{2} - 20 x + 100 = 0$ , 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).

2, 5
3, 6
20, 2
13, 1
4, 4
0, 9
$\frac{3}{4}, \frac{1}{4}$
$\frac{5}{8}, \frac{5}{7}$
$\frac{1}{2}, \frac{1}{3}$
$\frac{1}{2}, \frac{2}{3}$
± 5
± 1
$\pm \frac{1}{5}$
$\pm \sqrt{7}$
$\pm \sqrt{11}$
$\pm 2 \sqrt{3}$
3, 5, 8
−4, 0, 4
−9, −6, −2
± 1, 5
± 2, ± 5
$\pm 2 \sqrt{3}, \pm \sqrt{5}$

Answer Key 10.8

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