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

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

Subtract $\sqrt{x + 2}$ from both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(2 - \sqrt{x + 2})}^{2}$ .

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

$x = (2 - \sqrt{x + 2}) (2 - \sqrt{x + 2})$

Expand $(2 - \sqrt{x + 2}) (2 - \sqrt{x + 2})$ using the FOIL Method.

Apply the distributive property.

$x = 2 (2 - \sqrt{x + 2}) - \sqrt{x + 2} (2 - \sqrt{x + 2})$

Apply the distributive property.

$x = 2 \cdot 2 + 2 (- \sqrt{x + 2}) - \sqrt{x + 2} (2 - \sqrt{x + 2})$

Apply the distributive property.

$x = 2 \cdot 2 + 2 (- \sqrt{x + 2}) - \sqrt{x + 2} \cdot 2 - \sqrt{x + 2} (- \sqrt{x + 2})$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- \sqrt{x + 2} (- \sqrt{x + 2})$ .

Multiply $- 1$ by $- 1$ .

$x = 4 - 2 \sqrt{x + 2} - 2 \sqrt{x + 2} + 1 \sqrt{x + 2} \sqrt{x + 2}$

Multiply $\sqrt{x + 2}$ by $1$ .

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

Raise $\sqrt{x + 2}$ to the power of $1$ .

$x = 4 - 2 \sqrt{x + 2} - 2 \sqrt{x + 2} + {\sqrt{x + 2}}^{1} \sqrt{x + 2}$

Raise $\sqrt{x + 2}$ to the power of $1$ .

$x = 4 - 2 \sqrt{x + 2} - 2 \sqrt{x + 2} + {\sqrt{x + 2}}^{1} {\sqrt{x + 2}}^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 4 - 2 \sqrt{x + 2} - 2 \sqrt{x + 2} + {\sqrt{x + 2}}^{1 + 1}$

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Add $4$ and $2$ .

$x = 6 - 2 \sqrt{x + 2} - 2 \sqrt{x + 2} + x$

Subtract $2 \sqrt{x + 2}$ from $- 2 \sqrt{x + 2}$ .

$x = 6 - 4 \sqrt{x + 2} + x$

Solve for $- 4 \sqrt{x + 2}$ .

Rewrite the equation as $6 - 4 \sqrt{x + 2} + x = x$ .

$6 - 4 \sqrt{x + 2} + x = x$

Move all terms not containing $- 4 \sqrt{x + 2}$ to the right side of the equation.

Subtract $6$ from both sides of the equation.

$- 4 \sqrt{x + 2} + x = x - 6$

Subtract $x$ from both sides of the equation.

$- 4 \sqrt{x + 2} = x - 6 - x$

Combine the opposite terms in $x - 6 - x$ .

Subtract $x$ from $x$ .

$- 4 \sqrt{x + 2} = 0 - 6$

Subtract $6$ from $0$ .

$- 4 \sqrt{x + 2} = - 6$

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

${(- 4 \sqrt{x + 2})}^{2} = {(- 6)}^{2}$

Simplify each side of the equation.

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

${(- 4 {(x + 2)}^{\frac{1}{2}})}^{2} = {(- 6)}^{2}$

Simplify the left side.

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

Apply the product rule to $- 4 {(x + 2)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(x + 2)}^{\frac{1}{2}})}^{2} = {(- 6)}^{2}$

Raise $- 4$ to the power of $2$ .

$16 {({(x + 2)}^{\frac{1}{2}})}^{2} = {(- 6)}^{2}$

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

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

$16 {(x + 2)}^{\frac{1}{2} \cdot 2} = {(- 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x + 2)}^{\frac{1}{2} \cdot 2} = {(- 6)}^{2}$

Rewrite the expression.

$16 {(x + 2)}^{1} = {(- 6)}^{2}$

Simplify.

$16 (x + 2) = {(- 6)}^{2}$

Apply the distributive property.

$16 x + 16 \cdot 2 = {(- 6)}^{2}$

Multiply $16$ by $2$ .

$16 x + 32 = {(- 6)}^{2}$

Simplify the right side.

Raise $- 6$ to the power of $2$ .

$16 x + 32 = 36$

Solve for $x$ .

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

Subtract $32$ from both sides of the equation.

$16 x = 36 - 32$

Subtract $32$ from $36$ .

$16 x = 4$

Divide each term in $16 x = 4$ by $16$ and simplify.

Divide each term in $16 x = 4$ by $16$ .

$\frac{16 x}{16} = \frac{4}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{4}{16}$

Divide $x$ by $1$ .

$x = \frac{4}{16}$

Simplify the right side.

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$x = \frac{4 (1)}{16}$

Cancel the common factors.

Factor $4$ out of $16$ .

$x = \frac{4 \cdot 1}{4 \cdot 4}$

Cancel the common factor.

$x = \frac{4 \cdot 1}{4 \cdot 4}$

Rewrite the expression.

$x = \frac{1}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}$

Decimal Form:

$x = 0.25$

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

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