# What is the value of |m| in root(3)(m+9)=3+root(3)(m-9) ?

Oct 2, 2016

$| m | = 4 \sqrt{5}$

#### Explanation:

We will make use of the expansion of the cube of a binomial

${\left(a + b\right)}^{3} = {a}^{3} + 3 {a}^{2} b + 3 a {b}^{2} + {b}^{3}$

as well as the quadratic formula

$a {x}^{2} + b x + c = 0 \implies x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Proceeding,

$\sqrt{m + 9} = 3 + \sqrt{m - 9}$

$\implies m + 9 = {\left(3 + \sqrt{m - 9}\right)}^{3}$

$\implies m + 9 = 27 + 27 \sqrt{m - 9} + 9 {\left(\sqrt{m - 9}\right)}^{2} + m - 9$

$\implies 9 {\left(\sqrt{m - 9}\right)}^{2} + 27 \sqrt{m - 9} + 9 = 0$

$\implies {\left(\sqrt{m - 9}\right)}^{2} + 3 \sqrt{m - 9} + 1 = 0$

$\implies \sqrt{m - 9} = \frac{- 3 \pm \sqrt{{\left(- 3\right)}^{2} - 4 \left(1\right) \left(1\right)}}{2 \left(1\right)}$

$\implies \sqrt{m - 9} = \frac{- 3 \pm \sqrt{5}}{2}$

$\implies 2 \sqrt{m - 9} = - 3 \pm \sqrt{5}$

$\implies 8 \left(m - 9\right) = {\left(- 3 \pm \sqrt{5}\right)}^{3}$

$\implies 8 m - 72 = - 27 \pm 27 \sqrt{5} - 45 \pm 15 \sqrt{5}$

$\implies 8 m - 72 = - 72 \pm 32 \sqrt{5}$

$\implies 8 m = \pm 32 \sqrt{5}$

$\implies m = \pm 4 \sqrt{5}$

$\therefore | m | = 4 \sqrt{5}$