Solve for x square root of x+1+1 = square root of x+3

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

Subtract $1$ from both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

${(x + 1)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 3} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 1)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 3} - 1)}^{2}$

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(\sqrt{x + 3} - 1)}^{2}$ .

Rewrite ${(\sqrt{x + 3} - 1)}^{2}$ as $(\sqrt{x + 3} - 1) (\sqrt{x + 3} - 1)$ .

$x + 1 = (\sqrt{x + 3} - 1) (\sqrt{x + 3} - 1)$

Expand $(\sqrt{x + 3} - 1) (\sqrt{x + 3} - 1)$ using the FOIL Method.

Apply the distributive property.

$x + 1 = \sqrt{x + 3} (\sqrt{x + 3} - 1) - 1 (\sqrt{x + 3} - 1)$

Apply the distributive property.

$x + 1 = \sqrt{x + 3} \sqrt{x + 3} + \sqrt{x + 3} \cdot - 1 - 1 (\sqrt{x + 3} - 1)$

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 3} \sqrt{x + 3}$ .

Raise $\sqrt{x + 3}$ to the power of $1$ .

$x + 1 = {\sqrt{x + 3}}^{1} \sqrt{x + 3} + \sqrt{x + 3} \cdot - 1 - 1 \sqrt{x + 3} - 1 \cdot - 1$

Raise $\sqrt{x + 3}$ to the power of $1$ .

$x + 1 = {\sqrt{x + 3}}^{1} {\sqrt{x + 3}}^{1} + \sqrt{x + 3} \cdot - 1 - 1 \sqrt{x + 3} - 1 \cdot - 1$

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

$x + 1 = {\sqrt{x + 3}}^{1 + 1} + \sqrt{x + 3} \cdot - 1 - 1 \sqrt{x + 3} - 1 \cdot - 1$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x + 3}}^{2}$ as $x + 3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

$x + 1 = {({(x + 3)}^{\frac{1}{2}})}^{2} + \sqrt{x + 3} \cdot - 1 - 1 \sqrt{x + 3} - 1 \cdot - 1$

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

$x + 1 = {(x + 3)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 3} \cdot - 1 - 1 \sqrt{x + 3} - 1 \cdot - 1$

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

$x + 1 = {(x + 3)}^{\frac{2}{2}} + \sqrt{x + 3} \cdot - 1 - 1 \sqrt{x + 3} - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 1 = {(x + 3)}^{\frac{2}{2}} + \sqrt{x + 3} \cdot - 1 - 1 \sqrt{x + 3} - 1 \cdot - 1$

Rewrite the expression.

$x + 1 = {(x + 3)}^{1} + \sqrt{x + 3} \cdot - 1 - 1 \sqrt{x + 3} - 1 \cdot - 1$

Simplify.

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

Move $- 1$ to the left of $\sqrt{x + 3}$ .

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

Rewrite $- 1 \sqrt{x + 3}$ as $- \sqrt{x + 3}$ .

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

Rewrite $- 1 \sqrt{x + 3}$ as $- \sqrt{x + 3}$ .

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

Multiply $- 1$ by $- 1$ .

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

Add $3$ and $1$ .

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

Subtract $\sqrt{x + 3}$ from $- \sqrt{x + 3}$ .

$x + 1 = x + 4 - 2 \sqrt{x + 3}$

Solve for $- 2 \sqrt{x + 3}$ .

Rewrite the equation as $x + 4 - 2 \sqrt{x + 3} = x + 1$ .

$x + 4 - 2 \sqrt{x + 3} = x + 1$

Move all terms not containing $- 2 \sqrt{x + 3}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$4 - 2 \sqrt{x + 3} = x + 1 - x$

Subtract $4$ from both sides of the equation.

$- 2 \sqrt{x + 3} = x + 1 - x - 4$

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

Subtract $x$ from $x$ .

$- 2 \sqrt{x + 3} = 0 + 1 - 4$

Add $0$ and $1$ .

$- 2 \sqrt{x + 3} = 1 - 4$

Subtract $4$ from $1$ .

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

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

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

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${(- 2 {(x + 3)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 2 {(x + 3)}^{\frac{1}{2}}$ .

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

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

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

Multiply the exponents in ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $4$ by $3$ .

$4 x + 12 = {(- 3)}^{2}$

Simplify the right side.

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

$4 x + 12 = 9$

Solve for $x$ .

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

Subtract $12$ from both sides of the equation.

$4 x = 9 - 12$

Subtract $12$ from $9$ .

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{4}$

Decimal Form:

$x = - 0.75$

Solve for x square root of x+1+1 = square root of x+3

Related Questions

Support LCMGCF.com by disabling your adblocker.