Solve by Factoring square root of x+81=x-9

$\sqrt{x + 81} = x - 9$

Step 1

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

Subtract $x$ from both sides of the equation.

$\sqrt{x + 81} - x = - 9$

Add $9$ to both sides of the equation.

$\sqrt{x + 81} - x + 9 = 0$

Step 2

Move all terms not containing $\sqrt{x + 81}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{x + 81} + 9 = x$

Subtract $9$ from both sides of the equation.

$\sqrt{x + 81} = x - 9$

Step 3

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

${\sqrt{x + 81}}^{2} = {(x - 9)}^{2}$

Step 4

Simplify each side of the equation.

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

${({(x + 81)}^{\frac{1}{2}})}^{2} = {(x - 9)}^{2}$

Simplify the left side.

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

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

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

${(x + 81)}^{\frac{1}{2} \cdot 2} = {(x - 9)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 81)}^{\frac{1}{2} \cdot 2} = {(x - 9)}^{2}$

Rewrite the expression.

${(x + 81)}^{1} = {(x - 9)}^{2}$

Simplify.

$x + 81 = {(x - 9)}^{2}$

Simplify the right side.

Simplify ${(x - 9)}^{2}$ .

Rewrite ${(x - 9)}^{2}$ as $(x - 9) (x - 9)$ .

$x + 81 = (x - 9) (x - 9)$

Expand $(x - 9) (x - 9)$ using the FOIL Method.

Apply the distributive property.

$x + 81 = x (x - 9) - 9 (x - 9)$

Apply the distributive property.

$x + 81 = x \cdot x + x \cdot - 9 - 9 (x - 9)$

Apply the distributive property.

$x + 81 = x \cdot x + x \cdot - 9 - 9 x - 9 \cdot - 9$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 81 = x^{2} + x \cdot - 9 - 9 x - 9 \cdot - 9$

Move $- 9$ to the left of $x$ .

$x + 81 = x^{2} - 9 \cdot x - 9 x - 9 \cdot - 9$

Multiply $- 9$ by $- 9$ .

$x + 81 = x^{2} - 9 x - 9 x + 81$

Subtract $9 x$ from $- 9 x$ .

$x + 81 = x^{2} - 18 x + 81$

Step 5

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.

$x^{2} - 18 x + 81 = x + 81$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 18 x + 81 - x = 81$

Subtract $x$ from $- 18 x$ .

$x^{2} - 19 x + 81 = 81$

Subtract $81$ from both sides of the equation.

$x^{2} - 19 x + 81 - 81 = 0$

Combine the opposite terms in $x^{2} - 19 x + 81 - 81$ .

Subtract $81$ from $81$ .

$x^{2} - 19 x + 0 = 0$

Add $x^{2} - 19 x$ and $0$ .

$x^{2} - 19 x = 0$

Factor $x$ out of $x^{2} - 19 x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x - 19 x = 0$

Factor $x$ out of $- 19 x$ .

$x \cdot x + x \cdot - 19 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 19$ .

$x (x - 19) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 19 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 19$ equal to $0$ and solve for $x$ .

Set $x - 19$ equal to $0$ .

$x - 19 = 0$

Add $19$ to both sides of the equation.

$x = 19$

The final solution is all the values that make $x (x - 19) = 0$ true.

$x = 0, 19$

Step 6

Exclude the solutions that do not make $\sqrt{x + 81} - x + 9 = 0$ true.

$x = 19$

Solve by Factoring square root of x+81=x-9

Related Questions

Support LCMGCF.com by disabling your adblocker.