Solve for x x+3 = square root of x^2+5

$x + 3 = \sqrt{x^{2} + 5}$

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{x^{2} + 5} = x + 3$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} + 5}}^{2} = {(x + 3)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 5}$ as ${(x^{2} + 5)}^{\frac{1}{2}}$ .

${({(x^{2} + 5)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Simplify the left side.

Simplify ${({(x^{2} + 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} + 5)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 5)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} + 5)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Rewrite the expression.

${(x^{2} + 5)}^{1} = {(x + 3)}^{2}$

Simplify.

$x^{2} + 5 = {(x + 3)}^{2}$

Simplify the right side.

Simplify ${(x + 3)}^{2}$ .

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$x^{2} + 5 = (x + 3) (x + 3)$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + 5 = x (x + 3) + 3 (x + 3)$

Apply the distributive property.

$x^{2} + 5 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Apply the distributive property.

$x^{2} + 5 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + 5 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Move $3$ to the left of $x$ .

$x^{2} + 5 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$x^{2} + 5 = x^{2} + 3 x + 3 x + 9$

Add $3 x$ and $3 x$ .

$x^{2} + 5 = x^{2} + 6 x + 9$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 6 x + 9 = x^{2} + 5$

Move all terms containing $x$ to the left side of the equation.

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + 6 x + 9 - x^{2} = 5$

Combine the opposite terms in $x^{2} + 6 x + 9 - x^{2}$ .

Subtract $x^{2}$ from $x^{2}$ .

$6 x + 9 + 0 = 5$

Add $6 x + 9$ and $0$ .

$6 x + 9 = 5$

Move all terms not containing $x$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$6 x = 5 - 9$

Subtract $9$ from $5$ .

$6 x = - 4$

Divide each term in $6 x = - 4$ by $6$ and simplify.

Divide each term in $6 x = - 4$ by $6$ .

$\frac{6 x}{6} = \frac{- 4}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 4}{6}$

Divide $x$ by $1$ .

$x = \frac{- 4}{6}$

Simplify the right side.

Cancel the common factor of $- 4$ and $6$ .

Factor $2$ out of $- 4$ .

$x = \frac{2 (- 2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 2}{2 \cdot 3}$

Cancel the common factor.

$x = \frac{2 \cdot - 2}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{- 2}{3}$

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{3}$

Decimal Form:

$x = - 0.\overline{6}$

Solve for x x+3 = square root of x^2+5

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