# How do you solve sqrt(5x^2+11)=x+5 and identify any restrictions?

Mar 19, 2017

$x = - 1$ and $x = \frac{7}{2}$. This equation has no restrictions since $5 {x}^{2} + 11 \ge 0$.

#### Explanation:

Begin by squaring both sides of the equation to eliminate the square root. Recall that in squaring the right hand side, we can't square each term individually--we have to square the entire right hand side.

${\left(\sqrt{5 {x}^{2} + 11}\right)}^{2} = {\left(x + 5\right)}^{2}$

On the left, the sqrt and ""^2 cancel. On the right, expand the square by FOILing:

$5 {x}^{2} + 11 = \left(x + 5\right) \left(x + 5\right) = {x}^{2} + 5 x + 5 x + 25$

$5 {x}^{2} + 11 = {x}^{2} + 10 x + 25$

Putting all the terms on the same side:

$4 {x}^{2} - 10 x - 14 = 0$

Divide all terms by $2$:

$2 {x}^{2} - 5 x - 7 = 0$

Which we can factor by splitting the middle term:

$2 {x}^{2} + 2 x - 7 x - 7 = 0$

$2 x \left(x + 1\right) - 7 \left(x + 1\right) = 0$

$\left(2 x - 7\right) \left(x + 1\right) = 0$

Which give $x = \frac{7}{2}$ and $x = - 1$.

Check both of these by plugging them into the original equation:

Checking $x = \frac{7}{2}$:

$\sqrt{5 {\left(\frac{7}{2}\right)}^{2} + 11} = \frac{7}{2} + 5$

$\sqrt{5 \left(\frac{49}{4}\right) + 11} = \frac{7}{2} + \frac{10}{2}$

$\sqrt{\frac{245}{4} + \frac{44}{4}} = \frac{7}{2} + \frac{10}{2}$

$\sqrt{\frac{289}{4}} = \frac{17}{2}$

Which is true! So $x = \frac{7}{2}$ is a valid solution.

Checking $x = - 1$:

$\sqrt{5 {\left(- 1\right)}^{2} + 11} = - 1 + 5$

$\sqrt{5 + 11} = 4$

$\sqrt{16} = 4$

Which is true as well, so our solutions are $x = - 1$ and $x = \frac{7}{2}$.

We can skip the process of going back and checking answers by noting that since we have $\sqrt{5 {x}^{2} + 11}$ in the original equation, we can't have values of $x$ where $5 {x}^{2} + 11 < 0$, since we can't take the square root of a negative value.

Note that since $5 {x}^{2}$ and $11$ are both always positive, $5 {x}^{2} + 11$ will also always be positive. Thus, there is never a time when it's negative and by the same coin there is never a time when $\sqrt{5 {x}^{2} + 11}$ won't be defined. Thus there are no restrictions on this equation.