Solve by Factoring square root of x-5=5- square root of x

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $5$ from both sides of the equation.

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

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

$\sqrt{x - 5} - 5 + \sqrt{x} = 0$

Step 2

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

Add $5$ to both sides of the equation.

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

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

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$x - 5 = (5 - \sqrt{x}) (5 - \sqrt{x})$

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

Apply the distributive property.

$x - 5 = 5 (5 - \sqrt{x}) - \sqrt{x} (5 - \sqrt{x})$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$x - 5 = 25 + 5 (- \sqrt{x}) - \sqrt{x} \cdot 5 - \sqrt{x} (- \sqrt{x})$

Multiply $- 1$ by $5$ .

$x - 5 = 25 - 5 \sqrt{x} - \sqrt{x} \cdot 5 - \sqrt{x} (- \sqrt{x})$

Multiply $5$ by $- 1$ .

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} - \sqrt{x} (- \sqrt{x})$

Multiply $- \sqrt{x} (- \sqrt{x})$ .

Multiply $- 1$ by $- 1$ .

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + 1 \sqrt{x} \sqrt{x}$

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

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + \sqrt{x} \sqrt{x}$

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

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

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

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

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

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Add $1$ and $1$ .

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + {\sqrt{x}}^{2}$

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

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

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

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

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

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

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + x^{\frac{2}{2}}$

Rewrite the expression.

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + x^{1}$

Simplify.

$x - 5 = 25 - 5 \sqrt{x} - 5 \sqrt{x} + x$

Subtract $5 \sqrt{x}$ from $- 5 \sqrt{x}$ .

$x - 5 = 25 - 10 \sqrt{x} + x$

Step 5

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

Rewrite the equation as $25 - 10 \sqrt{x} + x = x - 5$ .

$25 - 10 \sqrt{x} + x = x - 5$

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

Subtract $25$ from both sides of the equation.

$- 10 \sqrt{x} + x = x - 5 - 25$

Subtract $x$ from both sides of the equation.

$- 10 \sqrt{x} = x - 5 - 25 - x$

Combine the opposite terms in $x - 5 - 25 - x$ .

Subtract $x$ from $x$ .

$- 10 \sqrt{x} = 0 - 5 - 25$

Subtract $5$ from $0$ .

$- 10 \sqrt{x} = - 5 - 25$

Subtract $25$ from $- 5$ .

$- 10 \sqrt{x} = - 30$

Step 6

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

${(- 10 \sqrt{x})}^{2} = {(- 30)}^{2}$

Step 7

Simplify each side of the equation.

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

${(- 10 x^{\frac{1}{2}})}^{2} = {(- 30)}^{2}$

Simplify the left side.

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

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

${(- 10)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 30)}^{2}$

Raise $- 10$ to the power of $2$ .

$100 {(x^{\frac{1}{2}})}^{2} = {(- 30)}^{2}$

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

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

$100 x^{\frac{1}{2} \cdot 2} = {(- 30)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 x^{\frac{1}{2} \cdot 2} = {(- 30)}^{2}$

Rewrite the expression.

$100 x^{1} = {(- 30)}^{2}$

Simplify.

$100 x = {(- 30)}^{2}$

Simplify the right side.

Raise $- 30$ to the power of $2$ .

$100 x = 900$

Step 8

Divide each term in $100 x = 900$ by $100$ and simplify.

Divide each term in $100 x = 900$ by $100$ .

$\frac{100 x}{100} = \frac{900}{100}$

Simplify the left side.

Cancel the common factor of $100$ .

Cancel the common factor.

$\frac{100 x}{100} = \frac{900}{100}$

Divide $x$ by $1$ .

$x = \frac{900}{100}$

Simplify the right side.

Divide $900$ by $100$ .

$x = 9$

Solve by Factoring square root of x-5=5- square root of x

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