Solve for x square root of 3x+1- square root of x-1=2

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

Add $\sqrt{x - 1}$ to both sides of the equation.

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

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.

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

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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 $2$ .

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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.

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

Solve for $x$ .

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

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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.

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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.

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Simplify each term.

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

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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 $x - 1$ .

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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 $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Simplify.

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

Subtract $1$ from $4$ .

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

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

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

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

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

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

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Subtract $3$ from both sides of the equation.

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $3 x$ .

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

Subtract $3$ from $1$ .

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

Divide each term by $4$ and simplify.

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Divide each term in $4 \sqrt{x - 1} = 2 x - 2$ by $4$ .

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

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

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

Simplify each term.

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Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 x$ .

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

Cancel the common factors.

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Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 2$ and $4$ .

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Factor $2$ out of $- 2$ .

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

Cancel the common factors.

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Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Simplify each side of the equation.

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Solve for $x$ .

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

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $\frac{x}{2} \cdot \frac{x}{2}$ .

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Multiply $\frac{x}{2}$ and $\frac{x}{2}$ .

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

Raise $x$ to the power of $1$ .

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

Raise $x$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

Multiply $\frac{x}{2} (- \frac{1}{2})$ .

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Multiply $\frac{x}{2}$ and $\frac{1}{2}$ .

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

Multiply $2$ by $2$ .

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

Multiply $- \frac{1}{2} \cdot \frac{x}{2}$ .

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Multiply $\frac{x}{2}$ and $\frac{1}{2}$ .

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

Multiply $2$ by $2$ .

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

Multiply $- \frac{1}{2} (- \frac{1}{2})$ .

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Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{1}{2}$ by $1$ .

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

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

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

Multiply $2$ by $2$ .

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

Subtract $\frac{x}{4}$ from $- \frac{x}{4}$ .

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

Simplify each term.

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Cancel the common factor of $2$ .

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Factor $2$ out of $- 2$ .

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

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Rewrite $- 1 \frac{x}{2}$ as $- \frac{x}{2}$ .

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

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

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

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

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Subtract $x$ from both sides of the equation.

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

To write $- x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $- x$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Find the common denominator.

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Multiply $\frac{- x - x \cdot 2}{2}$ by $\frac{2}{2}$ .

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

Combine.

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $2$ by $- 1$ .

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

Subtract $2 x$ from $- x$ .

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

Multiply $2$ by $- 3$ .

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

Multiply both sides of the equation by $4$ .

$x^{2} - 6 x + 1 = - 1 \cdot 4$

Remove parentheses.

$x^{2} - 6 x + 1 = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$x^{2} - 6 x + 1 = - 4$

Move $4$ to the left side of the equation by adding it to both sides.

$x^{2} - 6 x + 1 + 4 = 0$

Add $1$ and $4$ .

$x^{2} - 6 x + 5 = 0$

Factor $x^{2} - 6 x + 5$ using the AC method.

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Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $5$ and whose sum is $- 6$ .

$- 5, - 1$

Write the factored form using these integers.

$(x - 5) (x - 1) = 0$

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

$x - 5 = 0$

$x - 1 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(x - 5) (x - 1) = 0$ true.

$x = 5, 1$

Solve for x square root of 3x+1- square root of x-1=2

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