Solve by Factoring square root of 3x-3+ square root of 2x+8+1=0

$\sqrt{3 x - 3} + \sqrt{2 x + 8} + 1 = 0$

Step 1

Factor $3$ out of $3 x - 3$ .

Factor $3$ out of $3 x$ .

$\sqrt{3 (x) - 3} + \sqrt{2 x + 8} + 1 = 0$

Factor $3$ out of $- 3$ .

$\sqrt{3 (x) + 3 (- 1)} + \sqrt{2 x + 8} + 1 = 0$

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

$\sqrt{3 (x - 1)} + \sqrt{2 x + 8} + 1 = 0$

Step 2

Factor $2$ out of $2 x + 8$ .

Factor $2$ out of $2 x$ .

$\sqrt{3 (x - 1)} + \sqrt{2 (x) + 8} + 1 = 0$

Factor $2$ out of $8$ .

$\sqrt{3 (x - 1)} + \sqrt{2 x + 2 \cdot 4} + 1 = 0$

Factor $2$ out of $2 x + 2 \cdot 4$ .

$\sqrt{3 (x - 1)} + \sqrt{2 (x + 4)} + 1 = 0$

Step 3

Move all terms not containing $\sqrt{3 (x - 1)}$ to the right side of the equation.

Subtract $\sqrt{2 (x + 4)}$ from both sides of the equation.

$\sqrt{3 (x - 1)} + 1 = - \sqrt{2 (x + 4)}$

Subtract $1$ from both sides of the equation.

$\sqrt{3 (x - 1)} = - \sqrt{2 (x + 4)} - 1$

Step 4

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

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

Step 5

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

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

Multiply.

Multiply $3$ by $- 1$ .

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

Simplify.

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

Simplify the right side.

Simplify ${(- \sqrt{2 (x + 4)} - 1)}^{2}$ .

Rewrite ${(- \sqrt{2 (x + 4)} - 1)}^{2}$ as $(- \sqrt{2 (x + 4)} - 1) (- \sqrt{2 (x + 4)} - 1)$ .

$3 x - 3 = (- \sqrt{2 (x + 4)} - 1) (- \sqrt{2 (x + 4)} - 1)$

Expand $(- \sqrt{2 (x + 4)} - 1) (- \sqrt{2 (x + 4)} - 1)$ using the FOIL Method.

Apply the distributive property.

$3 x - 3 = - \sqrt{2 (x + 4)} (- \sqrt{2 (x + 4)} - 1) - 1 (- \sqrt{2 (x + 4)} - 1)$

Apply the distributive property.

$3 x - 3 = - \sqrt{2 (x + 4)} (- \sqrt{2 (x + 4)}) - \sqrt{2 (x + 4)} \cdot - 1 - 1 (- \sqrt{2 (x + 4)} - 1)$

Apply the distributive property.

$3 x - 3 = - \sqrt{2 (x + 4)} (- \sqrt{2 (x + 4)}) - \sqrt{2 (x + 4)} \cdot - 1 - 1 (- \sqrt{2 (x + 4)}) - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{2 (x + 4)} (- \sqrt{2 (x + 4)})$ .

Multiply $- 1$ by $- 1$ .

$3 x - 3 = 1 \sqrt{2 (x + 4)} \sqrt{2 (x + 4)} - \sqrt{2 (x + 4)} \cdot - 1 - 1 (- \sqrt{2 (x + 4)}) - 1 \cdot - 1$

Multiply $\sqrt{2 (x + 4)}$ by $1$ .

$3 x - 3 = \sqrt{2 (x + 4)} \sqrt{2 (x + 4)} - \sqrt{2 (x + 4)} \cdot - 1 - 1 (- \sqrt{2 (x + 4)}) - 1 \cdot - 1$

Raise $\sqrt{2 (x + 4)}$ to the power of $1$ .

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

Raise $\sqrt{2 (x + 4)}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2 (x + 4)}}^{2}$ as $2 (x + 4)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (x + 4)}$ as ${(2 (x + 4))}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$3 x - 3 = 2 (x + 4) - \sqrt{2 (x + 4)} \cdot - 1 - 1 (- \sqrt{2 (x + 4)}) - 1 \cdot - 1$

Apply the distributive property.

$3 x - 3 = 2 x + 2 \cdot 4 - \sqrt{2 (x + 4)} \cdot - 1 - 1 (- \sqrt{2 (x + 4)}) - 1 \cdot - 1$

Multiply $2$ by $4$ .

$3 x - 3 = 2 x + 8 - \sqrt{2 (x + 4)} \cdot - 1 - 1 (- \sqrt{2 (x + 4)}) - 1 \cdot - 1$

Multiply $- \sqrt{2 (x + 4)} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$3 x - 3 = 2 x + 8 + 1 \sqrt{2 (x + 4)} - 1 (- \sqrt{2 (x + 4)}) - 1 \cdot - 1$

Multiply $\sqrt{2 (x + 4)}$ by $1$ .

$3 x - 3 = 2 x + 8 + \sqrt{2 (x + 4)} - 1 (- \sqrt{2 (x + 4)}) - 1 \cdot - 1$

Multiply $- 1 (- \sqrt{2 (x + 4)})$ .

Multiply $- 1$ by $- 1$ .

$3 x - 3 = 2 x + 8 + \sqrt{2 (x + 4)} + 1 \sqrt{2 (x + 4)} - 1 \cdot - 1$

Multiply $\sqrt{2 (x + 4)}$ by $1$ .

$3 x - 3 = 2 x + 8 + \sqrt{2 (x + 4)} + \sqrt{2 (x + 4)} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$3 x - 3 = 2 x + 8 + \sqrt{2 (x + 4)} + \sqrt{2 (x + 4)} + 1$

Add $8$ and $1$ .

$3 x - 3 = 2 x + 9 + \sqrt{2 (x + 4)} + \sqrt{2 (x + 4)}$

Add $\sqrt{2 (x + 4)}$ and $\sqrt{2 (x + 4)}$ .

$3 x - 3 = 2 x + 9 + 2 \sqrt{2 (x + 4)}$

Step 6

Solve for $2 \sqrt{2 (x + 4)}$ .

Rewrite the equation as $2 x + 9 + 2 \sqrt{2 (x + 4)} = 3 x - 3$ .

$2 x + 9 + 2 \sqrt{2 (x + 4)} = 3 x - 3$

Move all terms not containing $2 \sqrt{2 (x + 4)}$ to the right side of the equation.

Subtract $2 x$ from both sides of the equation.

$9 + 2 \sqrt{2 (x + 4)} = 3 x - 3 - 2 x$

Subtract $9$ from both sides of the equation.

$2 \sqrt{2 (x + 4)} = 3 x - 3 - 2 x - 9$

Subtract $2 x$ from $3 x$ .

$2 \sqrt{2 (x + 4)} = x - 3 - 9$

Subtract $9$ from $- 3$ .

$2 \sqrt{2 (x + 4)} = x - 12$

Step 7

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

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

Step 8

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (x + 4)}$ as ${(2 (x + 4))}^{\frac{1}{2}}$ .

${(2 {(2 (x + 4))}^{\frac{1}{2}})}^{2} = {(x - 12)}^{2}$

Simplify the left side.

Simplify ${(2 {(2 (x + 4))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(2 {(2 x + 2 \cdot 4)}^{\frac{1}{2}})}^{2} = {(x - 12)}^{2}$

Simplify the expression.

Multiply $2$ by $4$ .

${(2 {(2 x + 8)}^{\frac{1}{2}})}^{2} = {(x - 12)}^{2}$

Apply the product rule to $2 {(2 x + 8)}^{\frac{1}{2}}$ .

$2^{2} {({(2 x + 8)}^{\frac{1}{2}})}^{2} = {(x - 12)}^{2}$

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

$4 {({(2 x + 8)}^{\frac{1}{2}})}^{2} = {(x - 12)}^{2}$

Multiply the exponents in ${({(2 x + 8)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(2 x + 8)}^{\frac{1}{2} \cdot 2} = {(x - 12)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(2 x + 8)}^{\frac{1}{2} \cdot 2} = {(x - 12)}^{2}$

Rewrite the expression.

$4 {(2 x + 8)}^{1} = {(x - 12)}^{2}$

Simplify.

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

Apply the distributive property.

$4 (2 x) + 4 \cdot 8 = {(x - 12)}^{2}$

Multiply.

Multiply $2$ by $4$ .

$8 x + 4 \cdot 8 = {(x - 12)}^{2}$

Multiply $4$ by $8$ .

$8 x + 32 = {(x - 12)}^{2}$

Simplify the right side.

Simplify ${(x - 12)}^{2}$ .

Rewrite ${(x - 12)}^{2}$ as $(x - 12) (x - 12)$ .

$8 x + 32 = (x - 12) (x - 12)$

Expand $(x - 12) (x - 12)$ using the FOIL Method.

Apply the distributive property.

$8 x + 32 = x (x - 12) - 12 (x - 12)$

Apply the distributive property.

$8 x + 32 = x \cdot x + x \cdot - 12 - 12 (x - 12)$

Apply the distributive property.

$8 x + 32 = x \cdot x + x \cdot - 12 - 12 x - 12 \cdot - 12$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$8 x + 32 = x^{2} + x \cdot - 12 - 12 x - 12 \cdot - 12$

Move $- 12$ to the left of $x$ .

$8 x + 32 = x^{2} - 12 \cdot x - 12 x - 12 \cdot - 12$

Multiply $- 12$ by $- 12$ .

$8 x + 32 = x^{2} - 12 x - 12 x + 144$

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

$8 x + 32 = x^{2} - 24 x + 144$

Step 9

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.

$x^{2} - 24 x + 144 = 8 x + 32$

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

Subtract $8 x$ from both sides of the equation.

$x^{2} - 24 x + 144 - 8 x = 32$

Subtract $8 x$ from $- 24 x$ .

$x^{2} - 32 x + 144 = 32$

Subtract $32$ from both sides of the equation.

$x^{2} - 32 x + 144 - 32 = 0$

Subtract $32$ from $144$ .

$x^{2} - 32 x + 112 = 0$

Factor $x^{2} - 32 x + 112$ 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 $112$ and whose sum is $- 32$ .

$- 28, - 4$

Write the factored form using these integers.

$(x - 28) (x - 4) = 0$

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

$x - 28 = 0$

$x - 4 = 0$

Set $x - 28$ equal to $0$ and solve for $x$ .

Set $x - 28$ equal to $0$ .

$x - 28 = 0$

Add $28$ to both sides of the equation.

$x = 28$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(x - 28) (x - 4) = 0$ true.

$x = 28, 4$

Step 10

Exclude the solutions that do not make $\sqrt{3 (x - 1)} + \sqrt{2 (x + 4)} + 1 = 0$ true.

$No solution$

Solve by Factoring square root of 3x-3+ square root of 2x+8+1=0

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