Solve 2x-x1/2-1=0 using the cuadratic formula?

Jan 28, 2018

$x = 1$. See explanation.

Explanation:

Given $2 x - {x}^{\frac{1}{2}} - 1 = 0$, let $u = {x}^{\frac{1}{2}}$ and the equation becomes:

$2 {u}^{2} - u - 1 = 0$

Using the quadratic formula on this we have:

$u = \frac{- \left(- 1\right) \pm \sqrt{{\left(- 1\right)}^{2} - 4 \left(2\right) \left(- 1\right)}}{2 \left(2\right)}$

$u = \frac{1 \pm \sqrt{1 + 8}}{4}$

$u = \frac{1 \pm 3}{4}$

So, $u = 1$ or $u = - \frac{1}{2}$, where $u = {x}^{\frac{1}{2}}$, which means:

${x}^{\frac{1}{2}} = 1 \rightarrow x = 1$

${x}^{\frac{1}{2}} \ne - \frac{1}{2}$ (think of the range of ${x}^{\frac{1}{2}}$)

So the only solution is $x = 1$.