# How do you solve 2sqrt(x+2)-3=7?

Apr 29, 2016

$x = 23$

#### Explanation:

Given:

$2 \sqrt{x + 2} - 3 = 7$

Add $3$ to both sides to get:

$2 \sqrt{x + 2} = 10$

Divide both sides by $2$ to get:

$\sqrt{x + 2} = 5$

Square both sides (noting that this can introduce spurious solutions) to get:

$x + 2 = 25$

Subtract $2$ from both sides to get:

$x = 23$

Check:

$2 \sqrt{23 + 2} - 3 = 2 \sqrt{25} - 3 = \left(2 \cdot 5\right) - 3 = 10 - 3 = 7$

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Notes

Why the concern about squaring both sides of the equation?

When you square both sides of an equation, solutions of the resulting equation may not be solutions of the original, due to having different signs. The function $f \left(x\right) = {x}^{2}$ maps positive and negative values to the same value.

$\sqrt{x + 2} = - 5$

then squaring both sides would give:

$x + 2 = 25$

and hence:

$x = 23$

but we have $\sqrt{23 + 2} = 5 \ne - 5$