Solve for x square root of 3x+5 = square root of 6x+34

$\sqrt{3 x} + 5 = \sqrt{6 x + 34}$

Step 1

Solve for $\sqrt{3 x}$ .

Simplify $\sqrt{6 x + 34}$ .

Rewrite.

$\sqrt{3 x} + 5 = 0 + 0 + \sqrt{6 x + 34}$

Simplify by adding zeros.

$\sqrt{3 x} + 5 = \sqrt{6 x + 34}$

Factor $2$ out of $6 x + 34$ .

Factor $2$ out of $6 x$ .

$\sqrt{3 x} + 5 = \sqrt{2 (3 x) + 34}$

Factor $2$ out of $34$ .

$\sqrt{3 x} + 5 = \sqrt{2 (3 x) + 2 (17)}$

Factor $2$ out of $2 (3 x) + 2 (17)$ .

$\sqrt{3 x} + 5 = \sqrt{2 (3 x + 17)}$

Subtract $5$ from both sides of the equation.

$\sqrt{3 x} = \sqrt{2 (3 x + 17)} - 5$

Step 2

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

${\sqrt{3 x}}^{2} = {(\sqrt{2 (3 x + 17)} - 5)}^{2}$

Step 3

Simplify each side of the equation.

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

${({(3 x)}^{\frac{1}{2}})}^{2} = {(\sqrt{2 (3 x + 17)} - 5)}^{2}$

Simplify the left side.

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

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

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

${(3 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 (3 x + 17)} - 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 (3 x + 17)} - 5)}^{2}$

Rewrite the expression.

${(3 x)}^{1} = {(\sqrt{2 (3 x + 17)} - 5)}^{2}$

Simplify.

$3 x = {(\sqrt{2 (3 x + 17)} - 5)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{2 (3 x + 17)} - 5)}^{2}$ .

Rewrite ${(\sqrt{2 (3 x + 17)} - 5)}^{2}$ as $(\sqrt{2 (3 x + 17)} - 5) (\sqrt{2 (3 x + 17)} - 5)$ .

$3 x = (\sqrt{2 (3 x + 17)} - 5) (\sqrt{2 (3 x + 17)} - 5)$

Expand $(\sqrt{2 (3 x + 17)} - 5) (\sqrt{2 (3 x + 17)} - 5)$ using the FOIL Method.

Apply the distributive property.

$3 x = \sqrt{2 (3 x + 17)} (\sqrt{2 (3 x + 17)} - 5) - 5 (\sqrt{2 (3 x + 17)} - 5)$

Apply the distributive property.

$3 x = \sqrt{2 (3 x + 17)} \sqrt{2 (3 x + 17)} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 (\sqrt{2 (3 x + 17)} - 5)$

Apply the distributive property.

$3 x = \sqrt{2 (3 x + 17)} \sqrt{2 (3 x + 17)} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{2 (3 x + 17)} \sqrt{2 (3 x + 17)}$ .

Raise $\sqrt{2 (3 x + 17)}$ to the power of $1$ .

$3 x = {\sqrt{2 (3 x + 17)}}^{1} \sqrt{2 (3 x + 17)} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Raise $\sqrt{2 (3 x + 17)}$ to the power of $1$ .

$3 x = {\sqrt{2 (3 x + 17)}}^{1} {\sqrt{2 (3 x + 17)}}^{1} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

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

$3 x = {\sqrt{2 (3 x + 17)}}^{1 + 1} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Add $1$ and $1$ .

$3 x = {\sqrt{2 (3 x + 17)}}^{2} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Rewrite ${\sqrt{2 (3 x + 17)}}^{2}$ as $2 (3 x + 17)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (3 x + 17)}$ as ${(2 (3 x + 17))}^{\frac{1}{2}}$ .

$3 x = {({(2 (3 x + 17))}^{\frac{1}{2}})}^{2} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

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

$3 x = {(2 (3 x + 17))}^{\frac{1}{2} \cdot 2} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

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

$3 x = {(2 (3 x + 17))}^{\frac{2}{2}} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 x = {(2 (3 x + 17))}^{\frac{2}{2}} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Rewrite the expression.

$3 x = {(2 (3 x + 17))}^{1} + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Simplify.

$3 x = 2 (3 x + 17) + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Apply the distributive property.

$3 x = 2 (3 x) + 2 \cdot 17 + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Multiply $3$ by $2$ .

$3 x = 6 x + 2 \cdot 17 + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Multiply $2$ by $17$ .

$3 x = 6 x + 34 + \sqrt{2 (3 x + 17)} \cdot - 5 - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Move $- 5$ to the left of $\sqrt{2 (3 x + 17)}$ .

$3 x = 6 x + 34 - 5 \cdot \sqrt{2 (3 x + 17)} - 5 \sqrt{2 (3 x + 17)} - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$3 x = 6 x + 34 - 5 \sqrt{2 (3 x + 17)} - 5 \sqrt{2 (3 x + 17)} + 25$

Add $34$ and $25$ .

$3 x = 6 x + 59 - 5 \sqrt{2 (3 x + 17)} - 5 \sqrt{2 (3 x + 17)}$

Subtract $5 \sqrt{2 (3 x + 17)}$ from $- 5 \sqrt{2 (3 x + 17)}$ .

$3 x = 6 x + 59 - 10 \sqrt{2 (3 x + 17)}$

Step 4

Solve for $- 10 \sqrt{2 (3 x + 17)}$ .

Rewrite the equation as $6 x + 59 - 10 \sqrt{2 (3 x + 17)} = 3 x$ .

$6 x + 59 - 10 \sqrt{2 (3 x + 17)} = 3 x$

Move all terms not containing $- 10 \sqrt{2 (3 x + 17)}$ to the right side of the equation.

Subtract $6 x$ from both sides of the equation.

$59 - 10 \sqrt{2 (3 x + 17)} = 3 x - 6 x$

Subtract $59$ from both sides of the equation.

$- 10 \sqrt{2 (3 x + 17)} = 3 x - 6 x - 59$

Subtract $6 x$ from $3 x$ .

$- 10 \sqrt{2 (3 x + 17)} = - 3 x - 59$

Step 5

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

${(- 10 \sqrt{2 (3 x + 17)})}^{2} = {(- 3 x - 59)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (3 x + 17)}$ as ${(2 (3 x + 17))}^{\frac{1}{2}}$ .

${(- 10 {(2 (3 x + 17))}^{\frac{1}{2}})}^{2} = {(- 3 x - 59)}^{2}$

Simplify the left side.

Simplify ${(- 10 {(2 (3 x + 17))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(- 10 {(2 (3 x) + 2 \cdot 17)}^{\frac{1}{2}})}^{2} = {(- 3 x - 59)}^{2}$

Simplify the expression.

Multiply $3$ by $2$ .

${(- 10 {(6 x + 2 \cdot 17)}^{\frac{1}{2}})}^{2} = {(- 3 x - 59)}^{2}$

Multiply $2$ by $17$ .

${(- 10 {(6 x + 34)}^{\frac{1}{2}})}^{2} = {(- 3 x - 59)}^{2}$

Apply the product rule to $- 10 {(6 x + 34)}^{\frac{1}{2}}$ .

${(- 10)}^{2} {({(6 x + 34)}^{\frac{1}{2}})}^{2} = {(- 3 x - 59)}^{2}$

Raise $- 10$ to the power of $2$ .

$100 {({(6 x + 34)}^{\frac{1}{2}})}^{2} = {(- 3 x - 59)}^{2}$

Multiply the exponents in ${({(6 x + 34)}^{\frac{1}{2}})}^{2}$ .

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

$100 {(6 x + 34)}^{\frac{1}{2} \cdot 2} = {(- 3 x - 59)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 {(6 x + 34)}^{\frac{1}{2} \cdot 2} = {(- 3 x - 59)}^{2}$

Rewrite the expression.

$100 {(6 x + 34)}^{1} = {(- 3 x - 59)}^{2}$

Simplify.

$100 (6 x + 34) = {(- 3 x - 59)}^{2}$

Apply the distributive property.

$100 (6 x) + 100 \cdot 34 = {(- 3 x - 59)}^{2}$

Multiply.

Multiply $6$ by $100$ .

$600 x + 100 \cdot 34 = {(- 3 x - 59)}^{2}$

Multiply $100$ by $34$ .

$600 x + 3400 = {(- 3 x - 59)}^{2}$

Simplify the right side.

Simplify ${(- 3 x - 59)}^{2}$ .

Rewrite ${(- 3 x - 59)}^{2}$ as $(- 3 x - 59) (- 3 x - 59)$ .

$600 x + 3400 = (- 3 x - 59) (- 3 x - 59)$

Expand $(- 3 x - 59) (- 3 x - 59)$ using the FOIL Method.

Apply the distributive property.

$600 x + 3400 = - 3 x (- 3 x - 59) - 59 (- 3 x - 59)$

Apply the distributive property.

$600 x + 3400 = - 3 x (- 3 x) - 3 x \cdot - 59 - 59 (- 3 x - 59)$

Apply the distributive property.

$600 x + 3400 = - 3 x (- 3 x) - 3 x \cdot - 59 - 59 (- 3 x) - 59 \cdot - 59$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$600 x + 3400 = - 3 \cdot - 3 x \cdot x - 3 x \cdot - 59 - 59 (- 3 x) - 59 \cdot - 59$

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

Move $x$ .

$600 x + 3400 = - 3 \cdot - 3 (x \cdot x) - 3 x \cdot - 59 - 59 (- 3 x) - 59 \cdot - 59$

Multiply $x$ by $x$ .

$600 x + 3400 = - 3 \cdot - 3 x^{2} - 3 x \cdot - 59 - 59 (- 3 x) - 59 \cdot - 59$

Multiply $- 3$ by $- 3$ .

$600 x + 3400 = 9 x^{2} - 3 x \cdot - 59 - 59 (- 3 x) - 59 \cdot - 59$

Multiply $- 59$ by $- 3$ .

$600 x + 3400 = 9 x^{2} + 177 x - 59 (- 3 x) - 59 \cdot - 59$

Multiply $- 3$ by $- 59$ .

$600 x + 3400 = 9 x^{2} + 177 x + 177 x - 59 \cdot - 59$

Multiply $- 59$ by $- 59$ .

$600 x + 3400 = 9 x^{2} + 177 x + 177 x + 3481$

Add $177 x$ and $177 x$ .

$600 x + 3400 = 9 x^{2} + 354 x + 3481$

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.

$9 x^{2} + 354 x + 3481 = 600 x + 3400$

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

Subtract $600 x$ from both sides of the equation.

$9 x^{2} + 354 x + 3481 - 600 x = 3400$

Subtract $600 x$ from $354 x$ .

$9 x^{2} - 246 x + 3481 = 3400$

Subtract $3400$ from both sides of the equation.

$9 x^{2} - 246 x + 3481 - 3400 = 0$

Subtract $3400$ from $3481$ .

$9 x^{2} - 246 x + 81 = 0$

Factor the left side of the equation.

Factor $3$ out of $9 x^{2} - 246 x + 81$ .

Factor $3$ out of $9 x^{2}$ .

$3 (3 x^{2}) - 246 x + 81 = 0$

Factor $3$ out of $- 246 x$ .

$3 (3 x^{2}) + 3 (- 82 x) + 81 = 0$

Factor $3$ out of $81$ .

$3 (3 x^{2}) + 3 (- 82 x) + 3 (27) = 0$

Factor $3$ out of $3 (3 x^{2}) + 3 (- 82 x)$ .

$3 (3 x^{2} - 82 x) + 3 (27) = 0$

Factor $3$ out of $3 (3 x^{2} - 82 x) + 3 (27)$ .

$3 (3 x^{2} - 82 x + 27) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 27 = 81$ and whose sum is $b = - 82$ .

Factor $- 82$ out of $- 82 x$ .

$3 (3 x^{2} - 82 x + 27) = 0$

Rewrite $- 82$ as $- 1$ plus $- 81$

$3 (3 x^{2} + (- 1 - 81) x + 27) = 0$

Apply the distributive property.

$3 (3 x^{2} - 1 x - 81 x + 27) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$3 ((3 x^{2} - 1 x) - 81 x + 27) = 0$

Factor out the greatest common factor (GCF) from each group.

$3 (x (3 x - 1) - 27 (3 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$3 ((3 x - 1) (x - 27)) = 0$

Remove unnecessary parentheses.

$3 (3 x - 1) (x - 27) = 0$

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

$3 x - 1 = 0$

$x - 27 = 0$

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

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.

$x = 27$

The final solution is all the values that make $3 (3 x - 1) (x - 27) = 0$ true.

$x = \frac{1}{3}, 27$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}, 27$

Decimal Form:

$x = 0.\overline{3}, 27$

Solve for x square root of 3x+5 = square root of 6x+34

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