Solve for x square root of x+2=1- square root of 3x+7

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(1 - \sqrt{3 x + 7})}^{2}$ .

Rewrite ${(1 - \sqrt{3 x + 7})}^{2}$ as $(1 - \sqrt{3 x + 7}) (1 - \sqrt{3 x + 7})$ .

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

Expand $(1 - \sqrt{3 x + 7}) (1 - \sqrt{3 x + 7})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $- \sqrt{3 x + 7}$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $- \sqrt{3 x + 7} (- \sqrt{3 x + 7})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{3 x + 7}$ by $1$ .

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

Raise $\sqrt{3 x + 7}$ to the power of $1$ .

$x + 2 = 1 - \sqrt{3 x + 7} - \sqrt{3 x + 7} + {\sqrt{3 x + 7}}^{1} \sqrt{3 x + 7}$

Raise $\sqrt{3 x + 7}$ to the power of $1$ .

$x + 2 = 1 - \sqrt{3 x + 7} - \sqrt{3 x + 7} + {\sqrt{3 x + 7}}^{1} {\sqrt{3 x + 7}}^{1}$

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

$x + 2 = 1 - \sqrt{3 x + 7} - \sqrt{3 x + 7} + {\sqrt{3 x + 7}}^{1 + 1}$

Add $1$ and $1$ .

$x + 2 = 1 - \sqrt{3 x + 7} - \sqrt{3 x + 7} + {\sqrt{3 x + 7}}^{2}$

Rewrite ${\sqrt{3 x + 7}}^{2}$ as $3 x + 7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 7}$ as ${(3 x + 7)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Add $1$ and $7$ .

$x + 2 = 8 - \sqrt{3 x + 7} - \sqrt{3 x + 7} + 3 x$

Subtract $\sqrt{3 x + 7}$ from $- \sqrt{3 x + 7}$ .

$x + 2 = 8 - 2 \sqrt{3 x + 7} + 3 x$

Step 3

Solve for $- 2 \sqrt{3 x + 7}$ .

Rewrite the equation as $8 - 2 \sqrt{3 x + 7} + 3 x = x + 2$ .

$8 - 2 \sqrt{3 x + 7} + 3 x = x + 2$

Move all terms not containing $- 2 \sqrt{3 x + 7}$ to the right side of the equation.

Subtract $8$ from both sides of the equation.

$- 2 \sqrt{3 x + 7} + 3 x = x + 2 - 8$

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

$- 2 \sqrt{3 x + 7} = x + 2 - 8 - 3 x$

Subtract $3 x$ from $x$ .

$- 2 \sqrt{3 x + 7} = - 2 x + 2 - 8$

Subtract $8$ from $2$ .

$- 2 \sqrt{3 x + 7} = - 2 x - 6$

Step 4

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

${(- 2 \sqrt{3 x + 7})}^{2} = {(- 2 x - 6)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 7}$ as ${(3 x + 7)}^{\frac{1}{2}}$ .

${(- 2 {(3 x + 7)}^{\frac{1}{2}})}^{2} = {(- 2 x - 6)}^{2}$

Simplify the left side.

Simplify ${(- 2 {(3 x + 7)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 2 {(3 x + 7)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(3 x + 7)}^{\frac{1}{2}})}^{2} = {(- 2 x - 6)}^{2}$

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

$4 {({(3 x + 7)}^{\frac{1}{2}})}^{2} = {(- 2 x - 6)}^{2}$

Multiply the exponents in ${({(3 x + 7)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(3 x + 7)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(3 x + 7)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 6)}^{2}$

Rewrite the expression.

$4 {(3 x + 7)}^{1} = {(- 2 x - 6)}^{2}$

Simplify.

$4 (3 x + 7) = {(- 2 x - 6)}^{2}$

Apply the distributive property.

$4 (3 x) + 4 \cdot 7 = {(- 2 x - 6)}^{2}$

Multiply.

Multiply $3$ by $4$ .

$12 x + 4 \cdot 7 = {(- 2 x - 6)}^{2}$

Multiply $4$ by $7$ .

$12 x + 28 = {(- 2 x - 6)}^{2}$

Simplify the right side.

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

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

$12 x + 28 = (- 2 x - 6) (- 2 x - 6)$

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

Apply the distributive property.

$12 x + 28 = - 2 x (- 2 x - 6) - 6 (- 2 x - 6)$

Apply the distributive property.

$12 x + 28 = - 2 x (- 2 x) - 2 x \cdot - 6 - 6 (- 2 x - 6)$

Apply the distributive property.

$12 x + 28 = - 2 x (- 2 x) - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$12 x + 28 = - 2 \cdot - 2 x \cdot x - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

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

Move $x$ .

$12 x + 28 = - 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

Multiply $x$ by $x$ .

$12 x + 28 = - 2 \cdot - 2 x^{2} - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

Multiply $- 2$ by $- 2$ .

$12 x + 28 = 4 x^{2} - 2 x \cdot - 6 - 6 (- 2 x) - 6 \cdot - 6$

Multiply $- 6$ by $- 2$ .

$12 x + 28 = 4 x^{2} + 12 x - 6 (- 2 x) - 6 \cdot - 6$

Multiply $- 2$ by $- 6$ .

$12 x + 28 = 4 x^{2} + 12 x + 12 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$12 x + 28 = 4 x^{2} + 12 x + 12 x + 36$

Add $12 x$ and $12 x$ .

$12 x + 28 = 4 x^{2} + 24 x + 36$

Step 6

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.

$4 x^{2} + 24 x + 36 = 12 x + 28$

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

Subtract $12 x$ from both sides of the equation.

$4 x^{2} + 24 x + 36 - 12 x = 28$

Subtract $12 x$ from $24 x$ .

$4 x^{2} + 12 x + 36 = 28$

Subtract $28$ from both sides of the equation.

$4 x^{2} + 12 x + 36 - 28 = 0$

Subtract $28$ from $36$ .

$4 x^{2} + 12 x + 8 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} + 12 x + 8$ .

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

$4 (x^{2}) + 12 x + 8 = 0$

Factor $4$ out of $12 x$ .

$4 (x^{2}) + 4 (3 x) + 8 = 0$

Factor $4$ out of $8$ .

$4 x^{2} + 4 (3 x) + 4 \cdot 2 = 0$

Factor $4$ out of $4 x^{2} + 4 (3 x)$ .

$4 (x^{2} + 3 x) + 4 \cdot 2 = 0$

Factor $4$ out of $4 (x^{2} + 3 x) + 4 \cdot 2$ .

$4 (x^{2} + 3 x + 2) = 0$

Factor.

Factor $x^{2} + 3 x + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$4 ((x + 1) (x + 2)) = 0$

Remove unnecessary parentheses.

$4 (x + 1) (x + 2) = 0$

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

$x + 1 = 0$

$x + 2 = 0$

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.

$x = - 1$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $4 (x + 1) (x + 2) = 0$ true.

$x = - 1, - 2$

Step 7

Exclude the solutions that do not make $\sqrt{x + 2} = 1 - \sqrt{3 x + 7}$ true.

$x = - 2$

Solve for x square root of x+2=1- square root of 3x+7

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