Evaluate square root of x^2+4=x+2

$\sqrt{x^{2} + 4} = x + 2$

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

${\sqrt{x^{2} + 4}}^{2} = {(x + 2)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 4}$ as ${(x^{2} + 4)}^{\frac{1}{2}}$ .

${({(x^{2} + 4)}^{\frac{1}{2}})}^{2} = {(x + 2)}^{2}$

Simplify the left side.

Simplify ${({(x^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

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

${(x^{2} + 4)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} + 4)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Rewrite the expression.

${(x^{2} + 4)}^{1} = {(x + 2)}^{2}$

Simplify.

$x^{2} + 4 = {(x + 2)}^{2}$

Simplify the right side.

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

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

$x^{2} + 4 = (x + 2) (x + 2)$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + 4 = x (x + 2) + 2 (x + 2)$

Apply the distributive property.

$x^{2} + 4 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Apply the distributive property.

$x^{2} + 4 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + 4 = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Move $2$ to the left of $x$ .

$x^{2} + 4 = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$x^{2} + 4 = x^{2} + 2 x + 2 x + 4$

Add $2 x$ and $2 x$ .

$x^{2} + 4 = x^{2} + 4 x + 4$

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.

$x^{2} + 4 x + 4 = x^{2} + 4$

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

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

$x^{2} + 4 x + 4 - x^{2} = 4$

Combine the opposite terms in $x^{2} + 4 x + 4 - x^{2}$ .

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

$4 x + 4 + 0 = 4$

Add $4 x + 4$ and $0$ .

$4 x + 4 = 4$

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

Subtract $4$ from both sides of the equation.

$4 x = 4 - 4$

Subtract $4$ from $4$ .

$4 x = 0$

Divide each term in $4 x = 0$ by $4$ and simplify.

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$x = 0$

Evaluate square root of x^2+4=x+2

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