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

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

Subtract $\sqrt{x - 1}$ from both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- \sqrt{x - 1} (- \sqrt{x - 1})$ .

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Subtract $1$ from $4$ .

$x + 1 = 3 - 2 \sqrt{x - 1} - 2 \sqrt{x - 1} + x$

Subtract $2 \sqrt{x - 1}$ from $- 2 \sqrt{x - 1}$ .

$x + 1 = 3 - 4 \sqrt{x - 1} + x$

Solve for $- 4 \sqrt{x - 1}$ .

Rewrite the equation as $3 - 4 \sqrt{x - 1} + x = x + 1$ .

$3 - 4 \sqrt{x - 1} + x = x + 1$

Move all terms not containing $- 4 \sqrt{x - 1}$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$- 4 \sqrt{x - 1} + x = x + 1 - 3$

Subtract $x$ from both sides of the equation.

$- 4 \sqrt{x - 1} = x + 1 - 3 - x$

Combine the opposite terms in $x + 1 - 3 - x$ .

Subtract $x$ from $x$ .

$- 4 \sqrt{x - 1} = 0 + 1 - 3$

Add $0$ and $1$ .

$- 4 \sqrt{x - 1} = 1 - 3$

Subtract $3$ from $1$ .

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $16$ by $- 1$ .

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

Simplify the right side.

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

$16 x - 16 = 4$

Solve for $x$ .

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

Add $16$ to both sides of the equation.

$16 x = 4 + 16$

Add $4$ and $16$ .

$16 x = 20$

Divide each term in $16 x = 20$ by $16$ and simplify.

Divide each term in $16 x = 20$ by $16$ .

$\frac{16 x}{16} = \frac{20}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{20}{16}$

Divide $x$ by $1$ .

$x = \frac{20}{16}$

Simplify the right side.

Cancel the common factor of $20$ and $16$ .

Factor $4$ out of $20$ .

$x = \frac{4 (5)}{16}$

Cancel the common factors.

Factor $4$ out of $16$ .

$x = \frac{4 \cdot 5}{4 \cdot 4}$

Cancel the common factor.

$x = \frac{4 \cdot 5}{4 \cdot 4}$

Rewrite the expression.

$x = \frac{5}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{4}$

Decimal Form:

$x = 1.25$

Mixed Number Form:

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

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

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