Solve for x square root of 3 = square root of x+ square root of x-6

$\sqrt{3} = \sqrt{x} + \sqrt{x - 6}$

Rewrite the equation as $\sqrt{x} + \sqrt{x - 6} = \sqrt{3}$ .

$\sqrt{x} + \sqrt{x - 6} = \sqrt{3}$

Subtract $\sqrt{x - 6}$ from both sides of the equation.

$\sqrt{x} = \sqrt{3} - \sqrt{x - 6}$

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

${\sqrt{x}}^{2} = {(\sqrt{3} - \sqrt{x - 6})}^{2}$

Simplify each side of the equation.

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

${(x^{\frac{1}{2}})}^{2} = {(\sqrt{3} - \sqrt{x - 6})}^{2}$

Simplify the left side.

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

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

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

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{3} - \sqrt{x - 6})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\sqrt{3} - \sqrt{x - 6})}^{2}$

Rewrite the expression.

$x^{1} = {(\sqrt{3} - \sqrt{x - 6})}^{2}$

Simplify.

$x = {(\sqrt{3} - \sqrt{x - 6})}^{2}$

Simplify the right side.

Simplify ${(\sqrt{3} - \sqrt{x - 6})}^{2}$ .

Rewrite ${(\sqrt{3} - \sqrt{x - 6})}^{2}$ as $(\sqrt{3} - \sqrt{x - 6}) (\sqrt{3} - \sqrt{x - 6})$ .

$x = (\sqrt{3} - \sqrt{x - 6}) (\sqrt{3} - \sqrt{x - 6})$

Expand $(\sqrt{3} - \sqrt{x - 6}) (\sqrt{3} - \sqrt{x - 6})$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{3} (\sqrt{3} - \sqrt{x - 6}) - \sqrt{x - 6} (\sqrt{3} - \sqrt{x - 6})$

Apply the distributive property.

$x = \sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{x - 6}) - \sqrt{x - 6} (\sqrt{3} - \sqrt{x - 6})$

Apply the distributive property.

$x = \sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{x - 6}) - \sqrt{x - 6} \sqrt{3} - \sqrt{x - 6} (- \sqrt{x - 6})$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$x = \sqrt{3 \cdot 3} + \sqrt{3} (- \sqrt{x - 6}) - \sqrt{x - 6} \sqrt{3} - \sqrt{x - 6} (- \sqrt{x - 6})$

Multiply $3$ by $3$ .

$x = \sqrt{9} + \sqrt{3} (- \sqrt{x - 6}) - \sqrt{x - 6} \sqrt{3} - \sqrt{x - 6} (- \sqrt{x - 6})$

Rewrite $9$ as $3^{2}$ .

$x = \sqrt{3^{2}} + \sqrt{3} (- \sqrt{x - 6}) - \sqrt{x - 6} \sqrt{3} - \sqrt{x - 6} (- \sqrt{x - 6})$

Pull terms out from under the radical, assuming positive real numbers.

$x = 3 + \sqrt{3} (- \sqrt{x - 6}) - \sqrt{x - 6} \sqrt{3} - \sqrt{x - 6} (- \sqrt{x - 6})$

Combine using the product rule for radicals.

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{x - 6} \sqrt{3} - \sqrt{x - 6} (- \sqrt{x - 6})$

Combine using the product rule for radicals.

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} - \sqrt{x - 6} (- \sqrt{x - 6})$

Multiply $- \sqrt{x - 6} (- \sqrt{x - 6})$ .

Multiply $- 1$ by $- 1$ .

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + 1 \sqrt{x - 6} \sqrt{x - 6}$

Multiply $\sqrt{x - 6}$ by $1$ .

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + \sqrt{x - 6} \sqrt{x - 6}$

Raise $\sqrt{x - 6}$ to the power of $1$ .

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {\sqrt{x - 6}}^{1} \sqrt{x - 6}$

Raise $\sqrt{x - 6}$ to the power of $1$ .

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {\sqrt{x - 6}}^{1} {\sqrt{x - 6}}^{1}$

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

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {\sqrt{x - 6}}^{1 + 1}$

Add $1$ and $1$ .

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {\sqrt{x - 6}}^{2}$

Rewrite ${\sqrt{x - 6}}^{2}$ as $x - 6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 6}$ as ${(x - 6)}^{\frac{1}{2}}$ .

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {({(x - 6)}^{\frac{1}{2}})}^{2}$

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

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {(x - 6)}^{\frac{1}{2} \cdot 2}$

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

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {(x - 6)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {(x - 6)}^{\frac{2}{2}}$

Rewrite the expression.

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + {(x - 6)}^{1}$

Simplify.

$x = 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + x - 6$

Subtract $6$ from $3$ .

$x = - 3 - \sqrt{3 (x - 6)} - \sqrt{3 (x - 6)} + x$

Subtract $\sqrt{3 (x - 6)}$ from $- \sqrt{3 (x - 6)}$ .

$x = - 3 - 2 \sqrt{3 (x - 6)} + x$

Solve for $- 2 \sqrt{3 (x - 6)}$ .

Rewrite the equation as $- 3 - 2 \sqrt{3 (x - 6)} + x = x$ .

$- 3 - 2 \sqrt{3 (x - 6)} + x = x$

Move all terms not containing $- 2 \sqrt{3 (x - 6)}$ to the right side of the equation.

Add $3$ to both sides of the equation.

$- 2 \sqrt{3 (x - 6)} + x = x + 3$

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{3 (x - 6)} = x + 3 - x$

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

Subtract $x$ from $x$ .

$- 2 \sqrt{3 (x - 6)} = 0 + 3$

Add $0$ and $3$ .

$- 2 \sqrt{3 (x - 6)} = 3$

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

${(- 2 \sqrt{3 (x - 6)})}^{2} = 3^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (x - 6)}$ as ${(3 (x - 6))}^{\frac{1}{2}}$ .

${(- 2 {(3 (x - 6))}^{\frac{1}{2}})}^{2} = 3^{2}$

Simplify the left side.

Simplify ${(- 2 {(3 (x - 6))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(- 2 {(3 x + 3 \cdot - 6)}^{\frac{1}{2}})}^{2} = 3^{2}$

Simplify the expression.

Multiply $3$ by $- 6$ .

${(- 2 {(3 x - 18)}^{\frac{1}{2}})}^{2} = 3^{2}$

Apply the product rule to $- 2 {(3 x - 18)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(3 x - 18)}^{\frac{1}{2}})}^{2} = 3^{2}$

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

$4 {({(3 x - 18)}^{\frac{1}{2}})}^{2} = 3^{2}$

Multiply the exponents in ${({(3 x - 18)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(3 x - 18)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(3 x - 18)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Rewrite the expression.

$4 {(3 x - 18)}^{1} = 3^{2}$

Simplify.

$4 (3 x - 18) = 3^{2}$

Apply the distributive property.

$4 (3 x) + 4 \cdot - 18 = 3^{2}$

Multiply.

Multiply $3$ by $4$ .

$12 x + 4 \cdot - 18 = 3^{2}$

Multiply $4$ by $- 18$ .

$12 x - 72 = 3^{2}$

Simplify the right side.

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

$12 x - 72 = 9$

Solve for $x$ .

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

Add $72$ to both sides of the equation.

$12 x = 9 + 72$

Add $9$ and $72$ .

$12 x = 81$

Divide each term in $12 x = 81$ by $12$ and simplify.

Divide each term in $12 x = 81$ by $12$ .

$\frac{12 x}{12} = \frac{81}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 x}{12} = \frac{81}{12}$

Divide $x$ by $1$ .

$x = \frac{81}{12}$

Simplify the right side.

Cancel the common factor of $81$ and $12$ .

Factor $3$ out of $81$ .

$x = \frac{3 (27)}{12}$

Cancel the common factors.

Factor $3$ out of $12$ .

$x = \frac{3 \cdot 27}{3 \cdot 4}$

Cancel the common factor.

$x = \frac{3 \cdot 27}{3 \cdot 4}$

Rewrite the expression.

$x = \frac{27}{4}$

Exclude the solutions that do not make $\sqrt{3} = \sqrt{x} + \sqrt{x - 6}$ true.

$No solution$

Solve for x square root of 3 = square root of x+ square root of x-6

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