Solve for x - square root of 6x+1-5=-7- square root of 3x+1

$- \sqrt{6 x + 1} - 5 = - 7 - \sqrt{3 x + 1}$

Step 1

Move all terms not containing $- \sqrt{6 x + 1}$ to the right side of the equation.

Add $5$ to both sides of the equation.

$- \sqrt{6 x + 1} = - 7 - \sqrt{3 x + 1} + 5$

Add $- 7$ and $5$ .

$- \sqrt{6 x + 1} = - 2 - \sqrt{3 x + 1}$

Step 2

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

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x + 1}$ as ${(6 x + 1)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

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

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

Raise $- 1$ to the power of $2$ .

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

Multiply ${({(6 x + 1)}^{\frac{1}{2}})}^{2}$ by $1$ .

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

Multiply the exponents in ${({(6 x + 1)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Rewrite ${(- 2 - \sqrt{3 x + 1})}^{2}$ as $(- 2 - \sqrt{3 x + 1}) (- 2 - \sqrt{3 x + 1})$ .

$6 x + 1 = (- 2 - \sqrt{3 x + 1}) (- 2 - \sqrt{3 x + 1})$

Expand $(- 2 - \sqrt{3 x + 1}) (- 2 - \sqrt{3 x + 1})$ using the FOIL Method.

Apply the distributive property.

$6 x + 1 = - 2 (- 2 - \sqrt{3 x + 1}) - \sqrt{3 x + 1} (- 2 - \sqrt{3 x + 1})$

Apply the distributive property.

$6 x + 1 = - 2 \cdot - 2 - 2 (- \sqrt{3 x + 1}) - \sqrt{3 x + 1} (- 2 - \sqrt{3 x + 1})$

Apply the distributive property.

$6 x + 1 = - 2 \cdot - 2 - 2 (- \sqrt{3 x + 1}) - \sqrt{3 x + 1} \cdot - 2 - \sqrt{3 x + 1} (- \sqrt{3 x + 1})$

Simplify and combine like terms.

Simplify each term.

Multiply $- 2$ by $- 2$ .

$6 x + 1 = 4 - 2 (- \sqrt{3 x + 1}) - \sqrt{3 x + 1} \cdot - 2 - \sqrt{3 x + 1} (- \sqrt{3 x + 1})$

Multiply $- 1$ by $- 2$ .

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} - \sqrt{3 x + 1} \cdot - 2 - \sqrt{3 x + 1} (- \sqrt{3 x + 1})$

Multiply $- 2$ by $- 1$ .

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} - \sqrt{3 x + 1} (- \sqrt{3 x + 1})$

Multiply $- \sqrt{3 x + 1} (- \sqrt{3 x + 1})$ .

Multiply $- 1$ by $- 1$ .

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + 1 \sqrt{3 x + 1} \sqrt{3 x + 1}$

Multiply $\sqrt{3 x + 1}$ by $1$ .

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + \sqrt{3 x + 1} \sqrt{3 x + 1}$

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

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {\sqrt{3 x + 1}}^{1} \sqrt{3 x + 1}$

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

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {\sqrt{3 x + 1}}^{1} {\sqrt{3 x + 1}}^{1}$

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

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {\sqrt{3 x + 1}}^{1 + 1}$

Add $1$ and $1$ .

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {\sqrt{3 x + 1}}^{2}$

Rewrite ${\sqrt{3 x + 1}}^{2}$ as $3 x + 1$ .

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

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {({(3 x + 1)}^{\frac{1}{2}})}^{2}$

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

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {(3 x + 1)}^{\frac{1}{2} \cdot 2}$

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

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {(3 x + 1)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {(3 x + 1)}^{\frac{2}{2}}$

Rewrite the expression.

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + {(3 x + 1)}^{1}$

Simplify.

$6 x + 1 = 4 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + 3 x + 1$

Add $4$ and $1$ .

$6 x + 1 = 5 + 2 \sqrt{3 x + 1} + 2 \sqrt{3 x + 1} + 3 x$

Add $2 \sqrt{3 x + 1}$ and $2 \sqrt{3 x + 1}$ .

$6 x + 1 = 5 + 4 \sqrt{3 x + 1} + 3 x$

Step 4

Solve for $4 \sqrt{3 x + 1}$ .

Rewrite the equation as $5 + 4 \sqrt{3 x + 1} + 3 x = 6 x + 1$ .

$5 + 4 \sqrt{3 x + 1} + 3 x = 6 x + 1$

Move all terms not containing $4 \sqrt{3 x + 1}$ to the right side of the equation.

Subtract $5$ from both sides of the equation.

$4 \sqrt{3 x + 1} + 3 x = 6 x + 1 - 5$

Subtract $3 x$ from both sides of the equation.

$4 \sqrt{3 x + 1} = 6 x + 1 - 5 - 3 x$

Subtract $3 x$ from $6 x$ .

$4 \sqrt{3 x + 1} = 3 x + 1 - 5$

Subtract $5$ from $1$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $4 {(3 x + 1)}^{\frac{1}{2}}$ .

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

Raise $4$ to the power of $2$ .

$16 {({(3 x + 1)}^{\frac{1}{2}})}^{2} = {(3 x - 4)}^{2}$

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

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

$16 {(3 x + 1)}^{\frac{1}{2} \cdot 2} = {(3 x - 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(3 x + 1)}^{\frac{1}{2} \cdot 2} = {(3 x - 4)}^{2}$

Rewrite the expression.

$16 {(3 x + 1)}^{1} = {(3 x - 4)}^{2}$

Simplify.

$16 (3 x + 1) = {(3 x - 4)}^{2}$

Apply the distributive property.

$16 (3 x) + 16 \cdot 1 = {(3 x - 4)}^{2}$

Multiply.

Multiply $3$ by $16$ .

$48 x + 16 \cdot 1 = {(3 x - 4)}^{2}$

Multiply $16$ by $1$ .

$48 x + 16 = {(3 x - 4)}^{2}$

Simplify the right side.

Simplify ${(3 x - 4)}^{2}$ .

Rewrite ${(3 x - 4)}^{2}$ as $(3 x - 4) (3 x - 4)$ .

$48 x + 16 = (3 x - 4) (3 x - 4)$

Expand $(3 x - 4) (3 x - 4)$ using the FOIL Method.

Apply the distributive property.

$48 x + 16 = 3 x (3 x - 4) - 4 (3 x - 4)$

Apply the distributive property.

$48 x + 16 = 3 x (3 x) + 3 x \cdot - 4 - 4 (3 x - 4)$

Apply the distributive property.

$48 x + 16 = 3 x (3 x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$48 x + 16 = 3 \cdot 3 x \cdot x + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

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

Move $x$ .

$48 x + 16 = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Multiply $x$ by $x$ .

$48 x + 16 = 3 \cdot 3 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Multiply $3$ by $3$ .

$48 x + 16 = 9 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Multiply $- 4$ by $3$ .

$48 x + 16 = 9 x^{2} - 12 x - 4 (3 x) - 4 \cdot - 4$

Multiply $3$ by $- 4$ .

$48 x + 16 = 9 x^{2} - 12 x - 12 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$48 x + 16 = 9 x^{2} - 12 x - 12 x + 16$

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

$48 x + 16 = 9 x^{2} - 24 x + 16$

Step 7

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.

$9 x^{2} - 24 x + 16 = 48 x + 16$

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

Subtract $48 x$ from both sides of the equation.

$9 x^{2} - 24 x + 16 - 48 x = 16$

Subtract $48 x$ from $- 24 x$ .

$9 x^{2} - 72 x + 16 = 16$

Subtract $16$ from both sides of the equation.

$9 x^{2} - 72 x + 16 - 16 = 0$

Combine the opposite terms in $9 x^{2} - 72 x + 16 - 16$ .

Subtract $16$ from $16$ .

$9 x^{2} - 72 x + 0 = 0$

Add $9 x^{2} - 72 x$ and $0$ .

$9 x^{2} - 72 x = 0$

Factor $9 x$ out of $9 x^{2} - 72 x$ .

Factor $9 x$ out of $9 x^{2}$ .

$9 x (x) - 72 x = 0$

Factor $9 x$ out of $- 72 x$ .

$9 x (x) + 9 x (- 8) = 0$

Factor $9 x$ out of $9 x (x) + 9 x (- 8)$ .

$9 x (x - 8) = 0$

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

$x = 0$

$x - 8 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

The final solution is all the values that make $9 x (x - 8) = 0$ true.

$x = 0, 8$

Step 8

Exclude the solutions that do not make $- \sqrt{6 x + 1} - 5 = - 7 - \sqrt{3 x + 1}$ true.

$x = 8$

Solve for x - square root of 6x+1-5=-7- square root of 3x+1

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