Solve for x square root of 6+x- square root of x-5=2

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

Step 1

Add $\sqrt{x - 5}$ to both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Move $2$ to the left of $\sqrt{x - 5}$ .

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

Multiply $\sqrt{x - 5} \sqrt{x - 5}$ .

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

$6 + x = 4 + 2 \sqrt{x - 5} + 2 \sqrt{x - 5} + {\sqrt{x - 5}}^{1} \sqrt{x - 5}$

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

$6 + x = 4 + 2 \sqrt{x - 5} + 2 \sqrt{x - 5} + {\sqrt{x - 5}}^{1} {\sqrt{x - 5}}^{1}$

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

$6 + x = 4 + 2 \sqrt{x - 5} + 2 \sqrt{x - 5} + {\sqrt{x - 5}}^{1 + 1}$

Add $1$ and $1$ .

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

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

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

$6 + x = 4 + 2 \sqrt{x - 5} + 2 \sqrt{x - 5} + {({(x - 5)}^{\frac{1}{2}})}^{2}$

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

$6 + x = 4 + 2 \sqrt{x - 5} + 2 \sqrt{x - 5} + {(x - 5)}^{\frac{1}{2} \cdot 2}$

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

$6 + x = 4 + 2 \sqrt{x - 5} + 2 \sqrt{x - 5} + {(x - 5)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$6 + x = 4 + 2 \sqrt{x - 5} + 2 \sqrt{x - 5} + {(x - 5)}^{\frac{2}{2}}$

Rewrite the expression.

$6 + x = 4 + 2 \sqrt{x - 5} + 2 \sqrt{x - 5} + {(x - 5)}^{1}$

Simplify.

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

Subtract $5$ from $4$ .

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

Add $2 \sqrt{x - 5}$ and $2 \sqrt{x - 5}$ .

$6 + x = - 1 + 4 \sqrt{x - 5} + x$

Step 4

Solve for $4 \sqrt{x - 5}$ .

Rewrite the equation as $- 1 + 4 \sqrt{x - 5} + x = 6 + x$ .

$- 1 + 4 \sqrt{x - 5} + x = 6 + x$

Move all terms not containing $4 \sqrt{x - 5}$ to the right side of the equation.

Add $1$ to both sides of the equation.

$4 \sqrt{x - 5} + x = 6 + x + 1$

Subtract $x$ from both sides of the equation.

$4 \sqrt{x - 5} = 6 + x + 1 - x$

Combine the opposite terms in $6 + x + 1 - x$ .

Subtract $x$ from $x$ .

$4 \sqrt{x - 5} = 6 + 0 + 1$

Add $6$ and $0$ .

$4 \sqrt{x - 5} = 6 + 1$

Add $6$ and $1$ .

$4 \sqrt{x - 5} = 7$

Step 5

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

${(4 \sqrt{x - 5})}^{2} = 7^{2}$

Step 6

Simplify each side of the equation.

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

${(4 {(x - 5)}^{\frac{1}{2}})}^{2} = 7^{2}$

Simplify the left side.

Simplify ${(4 {(x - 5)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 {(x - 5)}^{\frac{1}{2}}$ .

$4^{2} {({(x - 5)}^{\frac{1}{2}})}^{2} = 7^{2}$

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

$16 {({(x - 5)}^{\frac{1}{2}})}^{2} = 7^{2}$

Multiply the exponents in ${({(x - 5)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(x - 5)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x - 5)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Rewrite the expression.

$16 {(x - 5)}^{1} = 7^{2}$

Simplify.

$16 (x - 5) = 7^{2}$

Apply the distributive property.

$16 x + 16 \cdot - 5 = 7^{2}$

Multiply $16$ by $- 5$ .

$16 x - 80 = 7^{2}$

Simplify the right side.

Raise $7$ to the power of $2$ .

$16 x - 80 = 49$

Step 7

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Add $80$ to both sides of the equation.

$16 x = 49 + 80$

Add $49$ and $80$ .

$16 x = 129$

Divide each term in $16 x = 129$ by $16$ and simplify.

Divide each term in $16 x = 129$ by $16$ .

$\frac{16 x}{16} = \frac{129}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{129}{16}$

Divide $x$ by $1$ .

$x = \frac{129}{16}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{129}{16}$

Decimal Form:

$x = 8.0625$

Mixed Number Form:

$x = 8 \frac{1}{16}$

Solve for x square root of 6+x- square root of x-5=2

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