Solve for x square root of x+2+ square root of 3x+4=2

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

Step 1

Subtract $\sqrt{3 x + 4}$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

Expand $(2 - \sqrt{3 x + 4}) (2 - \sqrt{3 x + 4})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- \sqrt{3 x + 4} (- \sqrt{3 x + 4})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{3 x + 4}$ by $1$ .

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

Raise $\sqrt{3 x + 4}$ to the power of $1$ .

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

Raise $\sqrt{3 x + 4}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3 x + 4}}^{2}$ as $3 x + 4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Add $4$ and $4$ .

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

Subtract $2 \sqrt{3 x + 4}$ from $- 2 \sqrt{3 x + 4}$ .

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

Step 4

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

Rewrite the equation as $8 - 4 \sqrt{3 x + 4} + 3 x = x + 2$ .

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

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

Subtract $8$ from both sides of the equation.

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

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

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

Subtract $3 x$ from $x$ .

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

Subtract $8$ from $2$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

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

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

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

$16 {({(3 x + 4)}^{\frac{1}{2}})}^{2} = {(- 2 x - 6)}^{2}$

Multiply the exponents in ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(3 x + 4)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(3 x + 4)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 6)}^{2}$

Rewrite the expression.

$16 {(3 x + 4)}^{1} = {(- 2 x - 6)}^{2}$

Simplify.

$16 (3 x + 4) = {(- 2 x - 6)}^{2}$

Apply the distributive property.

$16 (3 x) + 16 \cdot 4 = {(- 2 x - 6)}^{2}$

Multiply.

Multiply $3$ by $16$ .

$48 x + 16 \cdot 4 = {(- 2 x - 6)}^{2}$

Multiply $16$ by $4$ .

$48 x + 64 = {(- 2 x - 6)}^{2}$

Simplify the right side.

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

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

$48 x + 64 = (- 2 x - 6) (- 2 x - 6)$

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

Apply the distributive property.

$48 x + 64 = - 2 x (- 2 x - 6) - 6 (- 2 x - 6)$

Apply the distributive property.

$48 x + 64 = - 2 x (- 2 x) - 2 x \cdot - 6 - 6 (- 2 x - 6)$

Apply the distributive property.

$48 x + 64 = - 2 x (- 2 x) - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$48 x + 64 = - 2 \cdot - 2 x \cdot x - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

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

Move $x$ .

$48 x + 64 = - 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

Multiply $x$ by $x$ .

$48 x + 64 = - 2 \cdot - 2 x^{2} - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

Multiply $- 2$ by $- 2$ .

$48 x + 64 = 4 x^{2} - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

Multiply $- 6$ by $- 2$ .

$48 x + 64 = 4 x^{2} + 12 x - 6 (- 2 x) - 6 \cdot - 6$

Multiply $- 2$ by $- 6$ .

$48 x + 64 = 4 x^{2} + 12 x + 12 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$48 x + 64 = 4 x^{2} + 12 x + 12 x + 36$

Add $12 x$ and $12 x$ .

$48 x + 64 = 4 x^{2} + 24 x + 36$

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.

$4 x^{2} + 24 x + 36 = 48 x + 64$

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

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

$4 x^{2} + 24 x + 36 - 48 x = 64$

Subtract $48 x$ from $24 x$ .

$4 x^{2} - 24 x + 36 = 64$

Subtract $64$ from both sides of the equation.

$4 x^{2} - 24 x + 36 - 64 = 0$

Subtract $64$ from $36$ .

$4 x^{2} - 24 x - 28 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 24 x - 28$ .

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

$4 (x^{2}) - 24 x - 28 = 0$

Factor $4$ out of $- 24 x$ .

$4 (x^{2}) + 4 (- 6 x) - 28 = 0$

Factor $4$ out of $- 28$ .

$4 x^{2} + 4 (- 6 x) + 4 \cdot - 7 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 6 x)$ .

$4 (x^{2} - 6 x) + 4 \cdot - 7 = 0$

Factor $4$ out of $4 (x^{2} - 6 x) + 4 \cdot - 7$ .

$4 (x^{2} - 6 x - 7) = 0$

Factor.

Factor $x^{2} - 6 x - 7$ 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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Write the factored form using these integers.

$4 ((x - 7) (x + 1)) = 0$

Remove unnecessary parentheses.

$4 (x - 7) (x + 1) = 0$

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

$x - 7 = 0$

$x + 1 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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 $4 (x - 7) (x + 1) = 0$ true.

$x = 7, - 1$

Step 8

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

$x = - 1$

Solve for x square root of x+2+ square root of 3x+4=2

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