Solve the Rational Equation for x square root of x+75 = square root of x-5

$\sqrt{x + 75} = \sqrt{x} - 5$

Step 1

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

${\sqrt{x + 75}}^{2} = {(\sqrt{x} - 5)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 75}$ as ${(x + 75)}^{\frac{1}{2}}$ .

${({(x + 75)}^{\frac{1}{2}})}^{2} = {(\sqrt{x} - 5)}^{2}$

Simplify the left side.

Simplify ${({(x + 75)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 75)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 75)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x} - 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 75)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x} - 5)}^{2}$

Rewrite the expression.

${(x + 75)}^{1} = {(\sqrt{x} - 5)}^{2}$

Simplify.

$x + 75 = {(\sqrt{x} - 5)}^{2}$

Simplify the right side.

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

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

$x + 75 = (\sqrt{x} - 5) (\sqrt{x} - 5)$

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

Apply the distributive property.

$x + 75 = \sqrt{x} (\sqrt{x} - 5) - 5 (\sqrt{x} - 5)$

Apply the distributive property.

$x + 75 = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 5 - 5 (\sqrt{x} - 5)$

Apply the distributive property.

$x + 75 = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

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

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

$x + 75 = {\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

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

$x + 75 = {\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

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

$x + 75 = {\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

Add $1$ and $1$ .

$x + 75 = {\sqrt{x}}^{2} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

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

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

$x + 75 = {(x^{\frac{1}{2}})}^{2} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

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

$x + 75 = x^{\frac{1}{2} \cdot 2} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

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

$x + 75 = x^{\frac{2}{2}} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 75 = x^{\frac{2}{2}} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

Rewrite the expression.

$x + 75 = x^{1} + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

Simplify.

$x + 75 = x + \sqrt{x} \cdot - 5 - 5 \sqrt{x} - 5 \cdot - 5$

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

$x + 75 = x - 5 \cdot \sqrt{x} - 5 \sqrt{x} - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$x + 75 = x - 5 \sqrt{x} - 5 \sqrt{x} + 25$

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

$x + 75 = x - 10 \sqrt{x} + 25$

Step 3

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

Rewrite the equation as $x - 10 \sqrt{x} + 25 = x + 75$ .

$x - 10 \sqrt{x} + 25 = x + 75$

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

Subtract $x$ from both sides of the equation.

$- 10 \sqrt{x} + 25 = x + 75 - x$

Subtract $25$ from both sides of the equation.

$- 10 \sqrt{x} = x + 75 - x - 25$

Combine the opposite terms in $x + 75 - x - 25$ .

Subtract $x$ from $x$ .

$- 10 \sqrt{x} = 0 + 75 - 25$

Add $0$ and $75$ .

$- 10 \sqrt{x} = 75 - 25$

Subtract $25$ from $75$ .

$- 10 \sqrt{x} = 50$

Step 4

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

${(- 10 \sqrt{x})}^{2} = 50^{2}$

Step 5

Simplify each side of the equation.

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

${(- 10 x^{\frac{1}{2}})}^{2} = 50^{2}$

Simplify the left side.

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

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

${(- 10)}^{2} {(x^{\frac{1}{2}})}^{2} = 50^{2}$

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

$100 {(x^{\frac{1}{2}})}^{2} = 50^{2}$

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

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

$100 x^{\frac{1}{2} \cdot 2} = 50^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 x^{\frac{1}{2} \cdot 2} = 50^{2}$

Rewrite the expression.

$100 x^{1} = 50^{2}$

Simplify.

$100 x = 50^{2}$

Simplify the right side.

Raise $50$ to the power of $2$ .

$100 x = 2500$

Step 6

Divide each term in $100 x = 2500$ by $100$ and simplify.

Divide each term in $100 x = 2500$ by $100$ .

$\frac{100 x}{100} = \frac{2500}{100}$

Simplify the left side.

Cancel the common factor of $100$ .

Cancel the common factor.

$\frac{100 x}{100} = \frac{2500}{100}$

Divide $x$ by $1$ .

$x = \frac{2500}{100}$

Simplify the right side.

Divide $2500$ by $100$ .

$x = 25$

Step 7

Exclude the solutions that do not make $\sqrt{x + 75} = \sqrt{x} - 5$ true.

$No solution$

Solve the Rational Equation for x square root of x+75 = square root of x-5

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