Solve by Factoring x-3 square root of x-2=2

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

Step 1

Subtract $2$ from both sides of the equation.

$x - 3 \sqrt{x - 2} - 2 = 0$

Step 2

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

Subtract $x$ from both sides of the equation.

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

Add $2$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

$9 {({(x - 2)}^{\frac{1}{2}})}^{2} = {(- x + 2)}^{2}$

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

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

$9 {(x - 2)}^{\frac{1}{2} \cdot 2} = {(- x + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 {(x - 2)}^{\frac{1}{2} \cdot 2} = {(- x + 2)}^{2}$

Rewrite the expression.

$9 {(x - 2)}^{1} = {(- x + 2)}^{2}$

Simplify.

$9 (x - 2) = {(- x + 2)}^{2}$

Apply the distributive property.

$9 x + 9 \cdot - 2 = {(- x + 2)}^{2}$

Multiply $9$ by $- 2$ .

$9 x - 18 = {(- x + 2)}^{2}$

Simplify the right side.

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

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

$9 x - 18 = (- x + 2) (- x + 2)$

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

Apply the distributive property.

$9 x - 18 = - x (- x + 2) + 2 (- x + 2)$

Apply the distributive property.

$9 x - 18 = - x (- x) - x \cdot 2 + 2 (- x + 2)$

Apply the distributive property.

$9 x - 18 = - x (- x) - x \cdot 2 + 2 (- x) + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$9 x - 18 = - 1 \cdot - 1 x \cdot x - x \cdot 2 + 2 (- x) + 2 \cdot 2$

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

Move $x$ .

$9 x - 18 = - 1 \cdot - 1 (x \cdot x) - x \cdot 2 + 2 (- x) + 2 \cdot 2$

Multiply $x$ by $x$ .

$9 x - 18 = - 1 \cdot - 1 x^{2} - x \cdot 2 + 2 (- x) + 2 \cdot 2$

Multiply $- 1$ by $- 1$ .

$9 x - 18 = 1 x^{2} - x \cdot 2 + 2 (- x) + 2 \cdot 2$

Multiply $x^{2}$ by $1$ .

$9 x - 18 = x^{2} - x \cdot 2 + 2 (- x) + 2 \cdot 2$

Multiply $2$ by $- 1$ .

$9 x - 18 = x^{2} - 2 x + 2 (- x) + 2 \cdot 2$

Multiply $- 1$ by $2$ .

$9 x - 18 = x^{2} - 2 x - 2 x + 2 \cdot 2$

Multiply $2$ by $2$ .

$9 x - 18 = x^{2} - 2 x - 2 x + 4$

Subtract $2 x$ from $- 2 x$ .

$9 x - 18 = x^{2} - 4 x + 4$

Step 5

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.

$x^{2} - 4 x + 4 = 9 x - 18$

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

Subtract $9 x$ from both sides of the equation.

$x^{2} - 4 x + 4 - 9 x = - 18$

Subtract $9 x$ from $- 4 x$ .

$x^{2} - 13 x + 4 = - 18$

Add $18$ to both sides of the equation.

$x^{2} - 13 x + 4 + 18 = 0$

Add $4$ and $18$ .

$x^{2} - 13 x + 22 = 0$

Factor $x^{2} - 13 x + 22$ 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 $22$ and whose sum is $- 13$ .

$- 11, - 2$

Write the factored form using these integers.

$(x - 11) (x - 2) = 0$

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

$x - 11 = 0$

$x - 2 = 0$

Set $x - 11$ equal to $0$ and solve for $x$ .

Set $x - 11$ equal to $0$ .

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 11) (x - 2) = 0$ true.

$x = 11, 2$

Solve by Factoring x-3 square root of x-2=2

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