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

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

Add $3$ to both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

Multiply $3$ by $3$ .

$3 x = x - 3 + 3 \sqrt{x - 3} + 3 \sqrt{x - 3} + 9$

Add $- 3$ and $9$ .

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

Add $3 \sqrt{x - 3}$ and $3 \sqrt{x - 3}$ .

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

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

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

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $6$ from both sides of the equation.

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

Subtract $x$ from $3 x$ .

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $6 {(x - 3)}^{\frac{1}{2}}$ .

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

Raise $6$ to the power of $2$ .

$36 {({(x - 3)}^{\frac{1}{2}})}^{2} = {(2 x - 6)}^{2}$

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

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

$36 {(x - 3)}^{\frac{1}{2} \cdot 2} = {(2 x - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(x - 3)}^{\frac{1}{2} \cdot 2} = {(2 x - 6)}^{2}$

Rewrite the expression.

$36 {(x - 3)}^{1} = {(2 x - 6)}^{2}$

Simplify.

$36 (x - 3) = {(2 x - 6)}^{2}$

Apply the distributive property.

$36 x + 36 \cdot - 3 = {(2 x - 6)}^{2}$

Multiply $36$ by $- 3$ .

$36 x - 108 = {(2 x - 6)}^{2}$

Simplify the right side.

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

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

$36 x - 108 = (2 x - 6) (2 x - 6)$

Expand $(2 x - 6) (2 x - 6)$ using the FOIL Method.

Apply the distributive property.

$36 x - 108 = 2 x (2 x - 6) - 6 (2 x - 6)$

Apply the distributive property.

$36 x - 108 = 2 x (2 x) + 2 x \cdot - 6 - 6 (2 x - 6)$

Apply the distributive property.

$36 x - 108 = 2 x (2 x) + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$36 x - 108 = 2 \cdot 2 x \cdot x + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6$

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

Move $x$ .

$36 x - 108 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6$

Multiply $x$ by $x$ .

$36 x - 108 = 2 \cdot 2 x^{2} + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6$

Multiply $2$ by $2$ .

$36 x - 108 = 4 x^{2} + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6$

Multiply $- 6$ by $2$ .

$36 x - 108 = 4 x^{2} - 12 x - 6 (2 x) - 6 \cdot - 6$

Multiply $2$ by $- 6$ .

$36 x - 108 = 4 x^{2} - 12 x - 12 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$36 x - 108 = 4 x^{2} - 12 x - 12 x + 36$

Subtract $12 x$ from $- 12 x$ .

$36 x - 108 = 4 x^{2} - 24 x + 36$

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.

$4 x^{2} - 24 x + 36 = 36 x - 108$

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

Subtract $36 x$ from both sides of the equation.

$4 x^{2} - 24 x + 36 - 36 x = - 108$

Subtract $36 x$ from $- 24 x$ .

$4 x^{2} - 60 x + 36 = - 108$

Add $108$ to both sides of the equation.

$4 x^{2} - 60 x + 36 + 108 = 0$

Add $36$ and $108$ .

$4 x^{2} - 60 x + 144 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 60 x + 144$ .

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

$4 (x^{2}) - 60 x + 144 = 0$

Factor $4$ out of $- 60 x$ .

$4 (x^{2}) + 4 (- 15 x) + 144 = 0$

Factor $4$ out of $144$ .

$4 x^{2} + 4 (- 15 x) + 4 \cdot 36 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 15 x)$ .

$4 (x^{2} - 15 x) + 4 \cdot 36 = 0$

Factor $4$ out of $4 (x^{2} - 15 x) + 4 \cdot 36$ .

$4 (x^{2} - 15 x + 36) = 0$

Factor.

Factor $x^{2} - 15 x + 36$ using the AC method.

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 $36$ and whose sum is $- 15$ .

$- 12, - 3$

Write the factored form using these integers.

$4 ((x - 12) (x - 3)) = 0$

Remove unnecessary parentheses.

$4 (x - 12) (x - 3) = 0$

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

$x - 12 = 0$

$x - 3 = 0$

Set $x - 12$ equal to $0$ and solve for $x$ .

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $4 (x - 12) (x - 3) = 0$ true.

$x = 12, 3$

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

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