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:

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

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.