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

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

Step 1

Subtract $\sqrt{2 x + 2}$ from both sides of the equation.

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

Step 2

Simplify each term.

Factor $3$ out of $3 x + 3$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $3$ .

$\sqrt{2 \sqrt{3 (x) + 3 (1)}} - \sqrt{2 x + 2} = 0$

Factor $3$ out of $3 (x) + 3 (1)$ .

$\sqrt{2 \sqrt{3 (x + 1)}} - \sqrt{2 x + 2} = 0$

Factor $2$ out of $2 x + 2$ .

Factor $2$ out of $2 x$ .

$\sqrt{2 \sqrt{3 (x + 1)}} - \sqrt{2 (x) + 2} = 0$

Factor $2$ out of $2$ .

$\sqrt{2 \sqrt{3 (x + 1)}} - \sqrt{2 (x) + 2 (1)} = 0$

Factor $2$ out of $2 (x) + 2 (1)$ .

$\sqrt{2 \sqrt{3 (x + 1)}} - \sqrt{2 (x + 1)} = 0$

Step 3

Add $\sqrt{2 (x + 1)}$ to both sides of the equation.

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

Step 4

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

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

Step 5

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Rewrite ${\sqrt{2 (x + 1)}}^{2}$ as $2 (x + 1)$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

Step 6

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

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

Step 7

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the distributive property.

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

Simplify the expression.

Multiply $3$ by $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$4 (3 x + 3) = {(2 x + 2)}^{2}$

Apply the distributive property.

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

Multiply.

Multiply $3$ by $4$ .

$12 x + 4 \cdot 3 = {(2 x + 2)}^{2}$

Multiply $4$ by $3$ .

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

Simplify the right side.

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

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

$12 x + 12 = (2 x + 2) (2 x + 2)$

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

Apply the distributive property.

$12 x + 12 = 2 x (2 x + 2) + 2 (2 x + 2)$

Apply the distributive property.

$12 x + 12 = 2 x (2 x) + 2 x \cdot 2 + 2 (2 x + 2)$

Apply the distributive property.

$12 x + 12 = 2 x (2 x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$12 x + 12 = 2 \cdot 2 x \cdot x + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

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

Move $x$ .

$12 x + 12 = 2 \cdot 2 (x \cdot x) + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

$12 x + 12 = 4 x^{2} + 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$

Multiply $2$ by $2$ .

$12 x + 12 = 4 x^{2} + 4 x + 2 (2 x) + 2 \cdot 2$

Multiply $2$ by $2$ .

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

Multiply $2$ by $2$ .

$12 x + 12 = 4 x^{2} + 4 x + 4 x + 4$

Add $4 x$ and $4 x$ .

$12 x + 12 = 4 x^{2} + 8 x + 4$

Step 8

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.

$4 x^{2} + 8 x + 4 = 12 x + 12$

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

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

$4 x^{2} + 8 x + 4 - 12 x = 12$

Subtract $12 x$ from $8 x$ .

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

Subtract $12$ from both sides of the equation.

$4 x^{2} - 4 x + 4 - 12 = 0$

Subtract $12$ from $4$ .

$4 x^{2} - 4 x - 8 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 4 x - 8$ .

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

$4 (x^{2}) - 4 x - 8 = 0$

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

$4 (x^{2}) + 4 (- x) - 8 = 0$

Factor $4$ out of $- 8$ .

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

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

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

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

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

Factor.

Factor $x^{2} - x - 2$ 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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

$4 ((x - 2) (x + 1)) = 0$

Remove unnecessary parentheses.

$4 (x - 2) (x + 1) = 0$

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

$x - 2 = 0$

$x + 1 = 0$

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.

$x = 2$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $4 (x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

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

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