Find All Complex Solutions square root of 3x+1=2+ square root of x+1

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Move $2$ to the left of $\sqrt{x + 1}$ .

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Add $4$ and $1$ .

$3 x + 1 = 5 + 2 \sqrt{x + 1} + 2 \sqrt{x + 1} + x$

Add $2 \sqrt{x + 1}$ and $2 \sqrt{x + 1}$ .

$3 x + 1 = 5 + 4 \sqrt{x + 1} + x$

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

Rewrite the equation as $5 + 4 \sqrt{x + 1} + x = 3 x + 1$ .

$5 + 4 \sqrt{x + 1} + x = 3 x + 1$

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

Subtract $5$ from both sides of the equation.

$4 \sqrt{x + 1} + x = 3 x + 1 - 5$

Subtract $x$ from both sides of the equation.

$4 \sqrt{x + 1} = 3 x + 1 - 5 - x$

Subtract $x$ from $3 x$ .

$4 \sqrt{x + 1} = 2 x + 1 - 5$

Subtract $5$ from $1$ .

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

$16 {({(x + 1)}^{\frac{1}{2}})}^{2} = {(2 x - 4)}^{2}$

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

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

$16 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(2 x - 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(2 x - 4)}^{2}$

Rewrite the expression.

$16 {(x + 1)}^{1} = {(2 x - 4)}^{2}$

Simplify.

$16 (x + 1) = {(2 x - 4)}^{2}$

Apply the distributive property.

$16 x + 16 \cdot 1 = {(2 x - 4)}^{2}$

Multiply $16$ by $1$ .

$16 x + 16 = {(2 x - 4)}^{2}$

Simplify the right side.

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

Rewrite ${(2 x - 4)}^{2}$ as $(2 x - 4) (2 x - 4)$ .

$16 x + 16 = (2 x - 4) (2 x - 4)$

Expand $(2 x - 4) (2 x - 4)$ using the FOIL Method.

Apply the distributive property.

$16 x + 16 = 2 x (2 x - 4) - 4 (2 x - 4)$

Apply the distributive property.

$16 x + 16 = 2 x (2 x) + 2 x \cdot - 4 - 4 (2 x - 4)$

Apply the distributive property.

$16 x + 16 = 2 x (2 x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$16 x + 16 = 2 \cdot 2 x \cdot x + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$16 x + 16 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4$

Multiply $x$ by $x$ .

$16 x + 16 = 2 \cdot 2 x^{2} + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4$

Multiply $2$ by $2$ .

$16 x + 16 = 4 x^{2} + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4$

Multiply $- 4$ by $2$ .

$16 x + 16 = 4 x^{2} - 8 x - 4 (2 x) - 4 \cdot - 4$

Multiply $2$ by $- 4$ .

$16 x + 16 = 4 x^{2} - 8 x - 8 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$16 x + 16 = 4 x^{2} - 8 x - 8 x + 16$

Subtract $8 x$ from $- 8 x$ .

$16 x + 16 = 4 x^{2} - 16 x + 16$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} - 16 x + 16 = 16 x + 16$

Move all terms containing $x$ to the left side of the equation.

Subtract $16 x$ from both sides of the equation.

$4 x^{2} - 16 x + 16 - 16 x = 16$

Subtract $16 x$ from $- 16 x$ .

$4 x^{2} - 32 x + 16 = 16$

Subtract $16$ from both sides of the equation.

$4 x^{2} - 32 x + 16 - 16 = 0$

Combine the opposite terms in $4 x^{2} - 32 x + 16 - 16$ .

Subtract $16$ from $16$ .

$4 x^{2} - 32 x + 0 = 0$

Add $4 x^{2} - 32 x$ and $0$ .

$4 x^{2} - 32 x = 0$

Factor $4 x$ out of $4 x^{2} - 32 x$ .

Factor $4 x$ out of $4 x^{2}$ .

$4 x (x) - 32 x = 0$

Factor $4 x$ out of $- 32 x$ .

$4 x (x) + 4 x (- 8) = 0$

Factor $4 x$ out of $4 x (x) + 4 x (- 8)$ .

$4 x (x - 8) = 0$

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

$x = 0$

$x - 8 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

The final solution is all the values that make $4 x (x - 8) = 0$ true.

$x = 0, 8$

Exclude the solutions that do not make $\sqrt{3 x + 1} = 2 + \sqrt{x + 1}$ true.

$x = 8$

Find All Complex Solutions square root of 3x+1=2+ square root of x+1

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