Solve for x square root of 6+2x=1+ square root of x+4

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Rewrite ${(1 + \sqrt{x + 4})}^{2}$ as $(1 + \sqrt{x + 4}) (1 + \sqrt{x + 4})$ .

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

Expand $(1 + \sqrt{x + 4}) (1 + \sqrt{x + 4})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

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

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

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

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

Multiply $\sqrt{x + 4} \sqrt{x + 4}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Add $1$ and $4$ .

$6 + 2 x = 5 + \sqrt{x + 4} + \sqrt{x + 4} + x$

Add $\sqrt{x + 4}$ and $\sqrt{x + 4}$ .

$6 + 2 x = 5 + 2 \sqrt{x + 4} + x$

Step 3

Solve for $2 \sqrt{x + 4}$ .

Rewrite the equation as $5 + 2 \sqrt{x + 4} + x = 6 + 2 x$ .

$5 + 2 \sqrt{x + 4} + x = 6 + 2 x$

Move all terms not containing $2 \sqrt{x + 4}$ to the right side of the equation.

Subtract $5$ from both sides of the equation.

$2 \sqrt{x + 4} + x = 6 + 2 x - 5$

Subtract $x$ from both sides of the equation.

$2 \sqrt{x + 4} = 6 + 2 x - 5 - x$

Subtract $5$ from $6$ .

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

Subtract $x$ from $2 x$ .

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

Step 4

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

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

Step 5

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $4$ by $4$ .

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

Simplify the right side.

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

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$4 x + 16 = (x + 1) (x + 1)$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$4 x + 16 = x (x + 1) + 1 (x + 1)$

Apply the distributive property.

$4 x + 16 = x \cdot x + x \cdot 1 + 1 (x + 1)$

Apply the distributive property.

$4 x + 16 = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$4 x + 16 = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$4 x + 16 = x^{2} + x + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$4 x + 16 = x^{2} + x + x + 1 \cdot 1$

Multiply $1$ by $1$ .

$4 x + 16 = x^{2} + x + x + 1$

Add $x$ and $x$ .

$4 x + 16 = x^{2} + 2 x + 1$

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.

$x^{2} + 2 x + 1 = 4 x + 16$

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

Subtract $4 x$ from both sides of the equation.

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

Subtract $4 x$ from $2 x$ .

$x^{2} - 2 x + 1 = 16$

Subtract $16$ from both sides of the equation.

$x^{2} - 2 x + 1 - 16 = 0$

Subtract $16$ from $1$ .

$x^{2} - 2 x - 15 = 0$

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$(x - 5) (x + 3) = 0$

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

$x - 5 = 0$

$x + 3 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(x - 5) (x + 3) = 0$ true.

$x = 5, - 3$

Step 7

Exclude the solutions that do not make $\sqrt{6 + 2 x} = 1 + \sqrt{x + 4}$ true.

$x = 5$

Solve for x square root of 6+2x=1+ square root of x+4

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