Solve by Factoring x-2 square root of 3x=9

$x - 2 \sqrt{3 x} = 9$

Step 1

Subtract $9$ from both sides of the equation.

$x - 2 \sqrt{3 x} - 9 = 0$

Step 2

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

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{3 x} - 9 = - x$

Add $9$ to both sides of the equation.

$- 2 \sqrt{3 x} = - x + 9$

Step 3

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

${(- 2 \sqrt{3 x})}^{2} = {(- x + 9)}^{2}$

Step 4

Simplify each side of the equation.

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

${(- 2 {(3 x)}^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

Simplify the left side.

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

Apply the product rule to $3 x$ .

${(- 2 (3^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = {(- x + 9)}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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

${(- 2 \cdot 3^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

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

${(- 2)}^{2} \cdot {(3^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

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

$4 \cdot {(3^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

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

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

$4 \cdot 3^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 \cdot 3^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

Rewrite the expression.

$4 \cdot 3^{1} {(x^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

Evaluate the exponent.

$4 \cdot 3 {(x^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

Multiply $4$ by $3$ .

$12 {(x^{\frac{1}{2}})}^{2} = {(- x + 9)}^{2}$

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

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

$12 x^{\frac{1}{2} \cdot 2} = {(- x + 9)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$12 x^{\frac{1}{2} \cdot 2} = {(- x + 9)}^{2}$

Rewrite the expression.

$12 x^{1} = {(- x + 9)}^{2}$

Simplify.

$12 x = {(- x + 9)}^{2}$

Simplify the right side.

Simplify ${(- x + 9)}^{2}$ .

Rewrite ${(- x + 9)}^{2}$ as $(- x + 9) (- x + 9)$ .

$12 x = (- x + 9) (- x + 9)$

Expand $(- x + 9) (- x + 9)$ using the FOIL Method.

Apply the distributive property.

$12 x = - x (- x + 9) + 9 (- x + 9)$

Apply the distributive property.

$12 x = - x (- x) - x \cdot 9 + 9 (- x + 9)$

Apply the distributive property.

$12 x = - x (- x) - x \cdot 9 + 9 (- x) + 9 \cdot 9$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$12 x = - 1 \cdot - 1 x \cdot x - x \cdot 9 + 9 (- x) + 9 \cdot 9$

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

Move $x$ .

$12 x = - 1 \cdot - 1 (x \cdot x) - x \cdot 9 + 9 (- x) + 9 \cdot 9$

Multiply $x$ by $x$ .

$12 x = - 1 \cdot - 1 x^{2} - x \cdot 9 + 9 (- x) + 9 \cdot 9$

Multiply $- 1$ by $- 1$ .

$12 x = 1 x^{2} - x \cdot 9 + 9 (- x) + 9 \cdot 9$

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

$12 x = x^{2} - x \cdot 9 + 9 (- x) + 9 \cdot 9$

Multiply $9$ by $- 1$ .

$12 x = x^{2} - 9 x + 9 (- x) + 9 \cdot 9$

Multiply $- 1$ by $9$ .

$12 x = x^{2} - 9 x - 9 x + 9 \cdot 9$

Multiply $9$ by $9$ .

$12 x = x^{2} - 9 x - 9 x + 81$

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

$12 x = x^{2} - 18 x + 81$

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.

$x^{2} - 18 x + 81 = 12 x$

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

Subtract $12 x$ from both sides of the equation.

$x^{2} - 18 x + 81 - 12 x = 0$

Subtract $12 x$ from $- 18 x$ .

$x^{2} - 30 x + 81 = 0$

Factor $x^{2} - 30 x + 81$ 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 $81$ and whose sum is $- 30$ .

$- 27, - 3$

Write the factored form using these integers.

$(x - 27) (x - 3) = 0$

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

$x - 27 = 0$

$x - 3 = 0$

Set $x - 27$ equal to $0$ and solve for $x$ .

Set $x - 27$ equal to $0$ .

$x - 27 = 0$

Add $27$ to both sides of the equation.

$x = 27$

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 $(x - 27) (x - 3) = 0$ true.

$x = 27, 3$

Step 6

Exclude the solutions that do not make $x - 2 \sqrt{3 x} - 9 = 0$ true.

$x = 27$

Solve by Factoring x-2 square root of 3x=9

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