# How do you solve 11 + sqrt(2x+1)=2?

## It was suggested I post this

Jan 16, 2018

No solutions.

#### Explanation:

Our goal is to isolate $x$ so we can find its value.

$11 + \sqrt{2 x + 1} = 2$

$\Rightarrow \sqrt{2 x + 1} = - 9$

We see that the equation has no solutions because the radical operator over $\mathbb{R}$ always returns positive values. So the equation has no solution.

Jan 16, 2018

#### Explanation:

Your goal here is to isolate the variable.

First, start by subtracting 2 from both sides, as such:

$11 - 2 + \sqrt{2 x + 1} = 2 - 2$

$9 + \sqrt{2 x + 1} = 0$

Then subtract 9 from both sides:

$9 - 9 + \sqrt{2 x + 1} = 0 - 9$

$\sqrt{2 x + 1} = - 9$

Now square both sides:

${\left(\sqrt{2 x + 1}\right)}^{2} = {\left(- 9\right)}^{2}$

$2 x + 1 = 81$

Then subtract 1 from both sides:

$2 x + 1 - 1 = 81 - 1$

$2 x = 80$

Finally, divided both sides by 2:

$\frac{2 x}{2} = \frac{80}{2}$

$x = 40$

Plug the answer into the original equation to check if it is correct:

$11 + \sqrt{2 \left(40\right) + 1} = 2$

$11 + 9 = 2$

$20 \ne 2$

When plugged in, we can see that $x = 40$ is not correct, so this problem has no solution.

Jan 16, 2018

No Real Solution

#### Explanation:

Begin by subtracting $11$ from both sides

$\cancel{11 \textcolor{red}{- 11}} + \sqrt{2 x + 1} = 2 \textcolor{red}{- 11}$

$\sqrt{2 x + 1} = - 9$

Square both sides to eliminate the radical on the left side

${\sqrt{2 x + 1}}^{\textcolor{red}{2}} = - {9}^{\textcolor{red}{2}}$

$2 x + 1 = 81$

Subtract $1$ from both sides

$2 x + \cancel{1 \textcolor{red}{- 1}} = 81 \textcolor{red}{- 1}$

$2 x = 80$

Divide both side by $2$

$\frac{\cancel{2}}{\cancel{\textcolor{red}{2}}} x = \frac{80}{\textcolor{red}{2}}$

$x = 40$

However if we verify this by substituting $40$ for $x$ in the original expression we'll find:

$11 + \sqrt{2 \left(\textcolor{red}{40}\right) + 1} = 2$

$11 + \sqrt{81} = 2$

$11 + 9 = 2$

$20 \ne 2$

$\therefore$ There are no real solutions

Jan 19, 2018

No solution for principle root $\to + \sqrt{2 x + 1} + 11 = 2$

The convention is; that unless the question states
$\pm \sqrt{\text{something}}$ we only use the $+ \sqrt{\text{smething}}$ which is the principle root.

#### Explanation:

However, just for fun; lets investigate the scenario of $\left(- 2\right) \times \left(- 2\right) = + 4 = \left(+ 2\right) \times \left(+ 2\right)$

So sqrt(4)=+-2" is definitely viable "ul("only if you ignore the convention")

The convention states :$\sqrt{4} \text{ can only "=+2" but you need"+-sqrt(4)" to have } \pm 2$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given: $\sqrt{2 x + 1} + 11 = 2$

Write as $\pm \sqrt{2 x + 1} + 11 = 2 \textcolor{red}{\leftarrow \text{ not the actual question}}$

Subtract 11 from both sides

$\pm \sqrt{2 x + 1} = - 9$

Clearly $+ \sqrt{2 x + 1} = - 9$ does not have a solution for $x \in \mathbb{R}$ as each side of the equals is the opposite sign.

However $- \sqrt{2 x + 1} = - 9$ may have a solution as both sides are negative. Accepting this should not be done as it is against the convention.$\textcolor{red}{\text{ Note that this is not the question as given}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Just for fun consider } - \sqrt{2 x + 1} = - 9}$

Square both sides

$2 x + 1 = + 81$

$2 x = 80$

$x = 40$
Which is not the the answer for the given question as written. The graph clearly shows no solution for $+ \sqrt{2 x + 1} = - 9$ which is the answer to the given question as written.

If the question had been $\pm \sqrt{2 x + 1}$ then we could legitimately have stated the answer as being: $\left(x , y\right) \to \left(40 , 9\right)$ is a solution for $- \sqrt{2 x + 1} = - 9$