Solve for x 2 square root of x+1+2 square root of x=1

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

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

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $4$ by $1$ .

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

Simplify the right side.

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

Rewrite ${(1 - 2 \sqrt{x})}^{2}$ as $(1 - 2 \sqrt{x}) (1 - 2 \sqrt{x})$ .

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

Expand $(1 - 2 \sqrt{x}) (1 - 2 \sqrt{x})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

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

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

Multiply $- 2$ by $1$ .

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

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

Multiply $- 2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

Rewrite the equation as $1 - 4 \sqrt{x} + 4 x = 4 x + 4$ .

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

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

Subtract $1$ from both sides of the equation.

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

Subtract $4 x$ from both sides of the equation.

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

Combine the opposite terms in $4 x + 4 - 1 - 4 x$ .

Subtract $4 x$ from $4 x$ .

$- 4 \sqrt{x} = 0 + 4 - 1$

Add $0$ and $4$ .

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

Subtract $1$ from $4$ .

$- 4 \sqrt{x} = 3$

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

${(- 4 \sqrt{x})}^{2} = 3^{2}$

Simplify each side of the equation.

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

${(- 4 x^{\frac{1}{2}})}^{2} = 3^{2}$

Simplify the left side.

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

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

${(- 4)}^{2} {(x^{\frac{1}{2}})}^{2} = 3^{2}$

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

$16 {(x^{\frac{1}{2}})}^{2} = 3^{2}$

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

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

$16 x^{\frac{1}{2} \cdot 2} = 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 x^{\frac{1}{2} \cdot 2} = 3^{2}$

Rewrite the expression.

$16 x^{1} = 3^{2}$

Simplify.

$16 x = 3^{2}$

Simplify the right side.

Raise $3$ to the power of $2$ .

$16 x = 9$

Divide each term in $16 x = 9$ by $16$ and simplify.

Divide each term in $16 x = 9$ by $16$ .

$\frac{16 x}{16} = \frac{9}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{9}{16}$

Divide $x$ by $1$ .

$x = \frac{9}{16}$

Exclude the solutions that do not make $2 \sqrt{x + 1} + 2 \sqrt{x} = 1$ true.

$No solution$

Solve for x 2 square root of x+1+2 square root of x=1

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