Solve the Rational Equation for x square root of x^2+81=x+10

$\sqrt{x^{2} + 81} = x + 10$

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

${\sqrt{x^{2} + 81}}^{2} = {(x + 10)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 81}$ as ${(x^{2} + 81)}^{\frac{1}{2}}$ .

${({(x^{2} + 81)}^{\frac{1}{2}})}^{2} = {(x + 10)}^{2}$

Simplify the left side.

Simplify ${({(x^{2} + 81)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} + 81)}^{\frac{1}{2}})}^{2}$ .

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

${(x^{2} + 81)}^{\frac{1}{2} \cdot 2} = {(x + 10)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} + 81)}^{\frac{1}{2} \cdot 2} = {(x + 10)}^{2}$

Rewrite the expression.

${(x^{2} + 81)}^{1} = {(x + 10)}^{2}$

Simplify.

$x^{2} + 81 = {(x + 10)}^{2}$

Simplify the right side.

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

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

$x^{2} + 81 = (x + 10) (x + 10)$

Expand $(x + 10) (x + 10)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + 81 = x (x + 10) + 10 (x + 10)$

Apply the distributive property.

$x^{2} + 81 = x \cdot x + x \cdot 10 + 10 (x + 10)$

Apply the distributive property.

$x^{2} + 81 = x \cdot x + x \cdot 10 + 10 x + 10 \cdot 10$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + 81 = x^{2} + x \cdot 10 + 10 x + 10 \cdot 10$

Move $10$ to the left of $x$ .

$x^{2} + 81 = x^{2} + 10 \cdot x + 10 x + 10 \cdot 10$

Multiply $10$ by $10$ .

$x^{2} + 81 = x^{2} + 10 x + 10 x + 100$

Add $10 x$ and $10 x$ .

$x^{2} + 81 = x^{2} + 20 x + 100$

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.

$x^{2} + 20 x + 100 = x^{2} + 81$

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

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + 20 x + 100 - x^{2} = 81$

Combine the opposite terms in $x^{2} + 20 x + 100 - x^{2}$ .

Subtract $x^{2}$ from $x^{2}$ .

$20 x + 100 + 0 = 81$

Add $20 x + 100$ and $0$ .

$20 x + 100 = 81$

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

Subtract $100$ from both sides of the equation.

$20 x = 81 - 100$

Subtract $100$ from $81$ .

$20 x = - 19$

Divide each term in $20 x = - 19$ by $20$ and simplify.

Divide each term in $20 x = - 19$ by $20$ .

$\frac{20 x}{20} = \frac{- 19}{20}$

Simplify the left side.

Cancel the common factor of $20$ .

Cancel the common factor.

$\frac{20 x}{20} = \frac{- 19}{20}$

Divide $x$ by $1$ .

$x = \frac{- 19}{20}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{19}{20}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{19}{20}$

Decimal Form:

$x = - 0.95$

Solve the Rational Equation for x square root of x^2+81=x+10

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