Solve for x square root of x+1+ square root of 2x = square root of 5x+3

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

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

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{5 x + 3} \sqrt{5 x + 3}$ .

Raise $\sqrt{5 x + 3}$ to the power of $1$ .

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

Raise $\sqrt{5 x + 3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{5 x + 3}}^{2}$ as $5 x + 3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 3}$ as ${(5 x + 3)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

Combine using the product rule for radicals.

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

Combine using the product rule for radicals.

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{2 x}$ by $1$ .

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

Raise $\sqrt{2 x}$ to the power of $1$ .

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

Raise $\sqrt{2 x}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Add $5 x$ and $2 x$ .

$x + 1 = 7 x + 3 - \sqrt{2 x (5 x + 3)} - \sqrt{(5 x + 3) \cdot 2 x}$

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

Move $5 x + 3$ .

$x + 1 = 7 x + 3 - \sqrt{2 x (5 x + 3)} - \sqrt{2 x (5 x + 3)}$

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

$x + 1 = 7 x + 3 - 2 \sqrt{2 x (5 x + 3)}$

Solve for $- 2 \sqrt{2 x (5 x + 3)}$ .

Rewrite the equation as $7 x + 3 - 2 \sqrt{2 x (5 x + 3)} = x + 1$ .

$7 x + 3 - 2 \sqrt{2 x (5 x + 3)} = x + 1$

Move all terms not containing $- 2 \sqrt{2 x (5 x + 3)}$ to the right side of the equation.

Subtract $7 x$ from both sides of the equation.

$3 - 2 \sqrt{2 x (5 x + 3)} = x + 1 - 7 x$

Subtract $3$ from both sides of the equation.

$- 2 \sqrt{2 x (5 x + 3)} = x + 1 - 7 x - 3$

Subtract $7 x$ from $x$ .

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

Subtract $3$ from $1$ .

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

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

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

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x (5 x + 3)}$ as ${(2 x (5 x + 3))}^{\frac{1}{2}}$ .

${(- 2 {(2 x (5 x + 3))}^{\frac{1}{2}})}^{2} = {(- 6 x - 2)}^{2}$

Simplify the left side.

Simplify ${(- 2 {(2 x (5 x + 3))}^{\frac{1}{2}})}^{2}$ .

Simplify by multiplying through.

Apply the distributive property.

${(- 2 {(2 x (5 x) + 2 x \cdot 3)}^{\frac{1}{2}})}^{2} = {(- 6 x - 2)}^{2}$

Simplify the expression.

Rewrite using the commutative property of multiplication.

${(- 2 {(2 \cdot 5 x \cdot x + 2 x \cdot 3)}^{\frac{1}{2}})}^{2} = {(- 6 x - 2)}^{2}$

Multiply $3$ by $2$ .

${(- 2 {(2 \cdot 5 x \cdot x + 6 x)}^{\frac{1}{2}})}^{2} = {(- 6 x - 2)}^{2}$

Simplify each term.

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

Move $x$ .

${(- 2 {(2 \cdot 5 (x \cdot x) + 6 x)}^{\frac{1}{2}})}^{2} = {(- 6 x - 2)}^{2}$

Multiply $x$ by $x$ .

${(- 2 {(2 \cdot 5 x^{2} + 6 x)}^{\frac{1}{2}})}^{2} = {(- 6 x - 2)}^{2}$

Multiply $2$ by $5$ .

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

Simplify the expression.

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

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

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

$4 {({(10 x^{2} + 6 x)}^{\frac{1}{2}})}^{2} = {(- 6 x - 2)}^{2}$

Multiply the exponents in ${({(10 x^{2} + 6 x)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(10 x^{2} + 6 x)}^{\frac{1}{2} \cdot 2} = {(- 6 x - 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(10 x^{2} + 6 x)}^{\frac{1}{2} \cdot 2} = {(- 6 x - 2)}^{2}$

Rewrite the expression.

$4 {(10 x^{2} + 6 x)}^{1} = {(- 6 x - 2)}^{2}$

Simplify.

$4 (10 x^{2} + 6 x) = {(- 6 x - 2)}^{2}$

Apply the distributive property.

$4 (10 x^{2}) + 4 (6 x) = {(- 6 x - 2)}^{2}$

Multiply.

Multiply $10$ by $4$ .

$40 x^{2} + 4 (6 x) = {(- 6 x - 2)}^{2}$

Multiply $6$ by $4$ .

$40 x^{2} + 24 x = {(- 6 x - 2)}^{2}$

Simplify the right side.

Simplify ${(- 6 x - 2)}^{2}$ .

Rewrite ${(- 6 x - 2)}^{2}$ as $(- 6 x - 2) (- 6 x - 2)$ .

$40 x^{2} + 24 x = (- 6 x - 2) (- 6 x - 2)$

Expand $(- 6 x - 2) (- 6 x - 2)$ using the FOIL Method.

Apply the distributive property.

$40 x^{2} + 24 x = - 6 x (- 6 x - 2) - 2 (- 6 x - 2)$

Apply the distributive property.

$40 x^{2} + 24 x = - 6 x (- 6 x) - 6 x \cdot - 2 - 2 (- 6 x - 2)$

Apply the distributive property.

$40 x^{2} + 24 x = - 6 x (- 6 x) - 6 x \cdot - 2 - 2 (- 6 x) - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$40 x^{2} + 24 x = - 6 \cdot - 6 x \cdot x - 6 x \cdot - 2 - 2 (- 6 x) - 2 \cdot - 2$

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

Move $x$ .

$40 x^{2} + 24 x = - 6 \cdot - 6 (x \cdot x) - 6 x \cdot - 2 - 2 (- 6 x) - 2 \cdot - 2$

Multiply $x$ by $x$ .

$40 x^{2} + 24 x = - 6 \cdot - 6 x^{2} - 6 x \cdot - 2 - 2 (- 6 x) - 2 \cdot - 2$

Multiply $- 6$ by $- 6$ .

$40 x^{2} + 24 x = 36 x^{2} - 6 x \cdot - 2 - 2 (- 6 x) - 2 \cdot - 2$

Multiply $- 2$ by $- 6$ .

$40 x^{2} + 24 x = 36 x^{2} + 12 x - 2 (- 6 x) - 2 \cdot - 2$

Multiply $- 6$ by $- 2$ .

$40 x^{2} + 24 x = 36 x^{2} + 12 x + 12 x - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$40 x^{2} + 24 x = 36 x^{2} + 12 x + 12 x + 4$

Add $12 x$ and $12 x$ .

$40 x^{2} + 24 x = 36 x^{2} + 24 x + 4$

Solve for $x$ .

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

Subtract $36 x^{2}$ from both sides of the equation.

$40 x^{2} + 24 x - 36 x^{2} = 24 x + 4$

Subtract $24 x$ from both sides of the equation.

$40 x^{2} + 24 x - 36 x^{2} - 24 x = 4$

Combine the opposite terms in $40 x^{2} + 24 x - 36 x^{2} - 24 x$ .

Subtract $24 x$ from $24 x$ .

$40 x^{2} - 36 x^{2} + 0 = 4$

Add $40 x^{2} - 36 x^{2}$ and $0$ .

$40 x^{2} - 36 x^{2} = 4$

Subtract $36 x^{2}$ from $40 x^{2}$ .

$4 x^{2} = 4$

Divide each term in $4 x^{2} = 4$ by $4$ and simplify.

Divide each term in $4 x^{2} = 4$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{4}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{4}{4}$

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

$x^{2} = \frac{4}{4}$

Simplify the right side.

Divide $4$ by $4$ .

$x^{2} = 1$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$x = \pm 1$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Exclude the solutions that do not make $\sqrt{x + 1} + \sqrt{2 x} = \sqrt{5 x + 3}$ true.

$x = 1$

Solve for x square root of x+1+ square root of 2x = square root of 5x+3

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