Solve the Rational Equation for x square root of x-2 = square root of x-10

$\sqrt{x} - 2 = \sqrt{x - 10}$

Step 1

Add $2$ to both sides of the equation.

$\sqrt{x} = \sqrt{x - 10} + 2$

Step 2

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

${\sqrt{x}}^{2} = {(\sqrt{x - 10} + 2)}^{2}$

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x = {(\sqrt{x - 10} + 2)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x - 10} + 2)}^{2}$ .

Rewrite ${(\sqrt{x - 10} + 2)}^{2}$ as $(\sqrt{x - 10} + 2) (\sqrt{x - 10} + 2)$ .

$x = (\sqrt{x - 10} + 2) (\sqrt{x - 10} + 2)$

Expand $(\sqrt{x - 10} + 2) (\sqrt{x - 10} + 2)$ using the FOIL Method.

Apply the distributive property.

$x = \sqrt{x - 10} (\sqrt{x - 10} + 2) + 2 (\sqrt{x - 10} + 2)$

Apply the distributive property.

$x = \sqrt{x - 10} \sqrt{x - 10} + \sqrt{x - 10} \cdot 2 + 2 (\sqrt{x - 10} + 2)$

Apply the distributive property.

$x = \sqrt{x - 10} \sqrt{x - 10} + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x - 10} \sqrt{x - 10}$ .

Raise $\sqrt{x - 10}$ to the power of $1$ .

$x = {\sqrt{x - 10}}^{1} \sqrt{x - 10} + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

Raise $\sqrt{x - 10}$ to the power of $1$ .

$x = {\sqrt{x - 10}}^{1} {\sqrt{x - 10}}^{1} + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

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

$x = {\sqrt{x - 10}}^{1 + 1} + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

Add $1$ and $1$ .

$x = {\sqrt{x - 10}}^{2} + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

Rewrite ${\sqrt{x - 10}}^{2}$ as $x - 10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 10}$ as ${(x - 10)}^{\frac{1}{2}}$ .

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

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

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

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

$x = {(x - 10)}^{\frac{2}{2}} + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = {(x - 10)}^{\frac{2}{2}} + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

Rewrite the expression.

$x = {(x - 10)}^{1} + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

Simplify.

$x = x - 10 + \sqrt{x - 10} \cdot 2 + 2 \sqrt{x - 10} + 2 \cdot 2$

Move $2$ to the left of $\sqrt{x - 10}$ .

$x = x - 10 + 2 \cdot \sqrt{x - 10} + 2 \sqrt{x - 10} + 2 \cdot 2$

Multiply $2$ by $2$ .

$x = x - 10 + 2 \sqrt{x - 10} + 2 \sqrt{x - 10} + 4$

Add $- 10$ and $4$ .

$x = x - 6 + 2 \sqrt{x - 10} + 2 \sqrt{x - 10}$

Add $2 \sqrt{x - 10}$ and $2 \sqrt{x - 10}$ .

$x = x - 6 + 4 \sqrt{x - 10}$

Step 4

Solve for $4 \sqrt{x - 10}$ .

Rewrite the equation as $x - 6 + 4 \sqrt{x - 10} = x$ .

$x - 6 + 4 \sqrt{x - 10} = x$

Move all terms not containing $4 \sqrt{x - 10}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- 6 + 4 \sqrt{x - 10} = x - x$

Add $6$ to both sides of the equation.

$4 \sqrt{x - 10} = x - x + 6$

Subtract $x$ from $x$ .

$4 \sqrt{x - 10} = 0 + 6$

Add $0$ and $6$ .

$4 \sqrt{x - 10} = 6$

Step 5

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

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

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 10}$ as ${(x - 10)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${(4 {(x - 10)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 {(x - 10)}^{\frac{1}{2}}$ .

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

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

$16 {({(x - 10)}^{\frac{1}{2}})}^{2} = 6^{2}$

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

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

$16 {(x - 10)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(x - 10)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Rewrite the expression.

$16 {(x - 10)}^{1} = 6^{2}$

Simplify.

$16 (x - 10) = 6^{2}$

Apply the distributive property.

$16 x + 16 \cdot - 10 = 6^{2}$

Multiply $16$ by $- 10$ .

$16 x - 160 = 6^{2}$

Simplify the right side.

Raise $6$ to the power of $2$ .

$16 x - 160 = 36$

Step 7

Solve for $x$ .

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

Add $160$ to both sides of the equation.

$16 x = 36 + 160$

Add $36$ and $160$ .

$16 x = 196$

Divide each term in $16 x = 196$ by $16$ and simplify.

Divide each term in $16 x = 196$ by $16$ .

$\frac{16 x}{16} = \frac{196}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x}{16} = \frac{196}{16}$

Divide $x$ by $1$ .

$x = \frac{196}{16}$

Simplify the right side.

Cancel the common factor of $196$ and $16$ .

Factor $4$ out of $196$ .

$x = \frac{4 (49)}{16}$

Cancel the common factors.

Factor $4$ out of $16$ .

$x = \frac{4 \cdot 49}{4 \cdot 4}$

Cancel the common factor.

$x = \frac{4 \cdot 49}{4 \cdot 4}$

Rewrite the expression.

$x = \frac{49}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{49}{4}$

Decimal Form:

$x = 12.25$

Mixed Number Form:

$x = 12 \frac{1}{4}$

Solve the Rational Equation for x square root of x-2 = square root of x-10

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