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

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Rewrite ${\sqrt{x^{2} + 5}}^{2}$ as $x^{2} + 5$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Solve for $x$ .

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

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

Subtract $5$ from both sides of the equation.

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

Combine the opposite terms in $x + 5 - x^{2} - 5$ .

Subtract $5$ from $5$ .

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

Add $x - x^{2}$ and $0$ .

$x - x^{2} = 0$

Factor $x$ out of $x - x^{2}$ .

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x^{2} = 0$

Factor $x$ out of $- x^{2}$ .

$x \cdot 1 + x (- x) = 0$

Factor $x$ out of $x \cdot 1 + x (- x)$ .

$x (1 - x) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $1 - x$ equal to $0$ and solve for $x$ .

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Solve $1 - x = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$- x = - 1$

Divide each term in $- x = - 1$ by $- 1$ and simplify.

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

The final solution is all the values that make $x (1 - x) = 0$ true.

$x = 0, 1$

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

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