Solve by Factoring square root of x-27=9- square root of x

$\sqrt{x - 27} = 9 - \sqrt{x}$

Step 1

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

Subtract $9$ from both sides of the equation.

$\sqrt{x - 27} - 9 = - \sqrt{x}$

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

$\sqrt{x - 27} - 9 + \sqrt{x} = 0$

Step 2

Move all terms not containing $\sqrt{x - 27}$ to the right side of the equation.

Add $9$ to both sides of the equation.

$\sqrt{x - 27} + \sqrt{x} = 9$

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

$\sqrt{x - 27} = 9 - \sqrt{x}$

Step 3

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

${\sqrt{x - 27}}^{2} = {(9 - \sqrt{x})}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 27}$ as ${(x - 27)}^{\frac{1}{2}}$ .

${({(x - 27)}^{\frac{1}{2}})}^{2} = {(9 - \sqrt{x})}^{2}$

Simplify the left side.

Simplify ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

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

${(x - 27)}^{\frac{1}{2} \cdot 2} = {(9 - \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 27)}^{\frac{1}{2} \cdot 2} = {(9 - \sqrt{x})}^{2}$

Rewrite the expression.

${(x - 27)}^{1} = {(9 - \sqrt{x})}^{2}$

Simplify.

$x - 27 = {(9 - \sqrt{x})}^{2}$

Simplify the right side.

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

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

$x - 27 = (9 - \sqrt{x}) (9 - \sqrt{x})$

Expand $(9 - \sqrt{x}) (9 - \sqrt{x})$ using the FOIL Method.

Apply the distributive property.

$x - 27 = 9 (9 - \sqrt{x}) - \sqrt{x} (9 - \sqrt{x})$

Apply the distributive property.

$x - 27 = 9 \cdot 9 + 9 (- \sqrt{x}) - \sqrt{x} (9 - \sqrt{x})$

Apply the distributive property.

$x - 27 = 9 \cdot 9 + 9 (- \sqrt{x}) - \sqrt{x} \cdot 9 - \sqrt{x} (- \sqrt{x})$

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $9$ .

$x - 27 = 81 + 9 (- \sqrt{x}) - \sqrt{x} \cdot 9 - \sqrt{x} (- \sqrt{x})$

Multiply $- 1$ by $9$ .

$x - 27 = 81 - 9 \sqrt{x} - \sqrt{x} \cdot 9 - \sqrt{x} (- \sqrt{x})$

Multiply $9$ by $- 1$ .

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} - \sqrt{x} (- \sqrt{x})$

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

Multiply $- 1$ by $- 1$ .

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + 1 \sqrt{x} \sqrt{x}$

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

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + \sqrt{x} \sqrt{x}$

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

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

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

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

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

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Add $1$ and $1$ .

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + {\sqrt{x}}^{2}$

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

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

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

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

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

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

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + x^{\frac{2}{2}}$

Rewrite the expression.

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + x^{1}$

Simplify.

$x - 27 = 81 - 9 \sqrt{x} - 9 \sqrt{x} + x$

Subtract $9 \sqrt{x}$ from $- 9 \sqrt{x}$ .

$x - 27 = 81 - 18 \sqrt{x} + x$

Step 5

Solve for $- 18 \sqrt{x}$ .

Rewrite the equation as $81 - 18 \sqrt{x} + x = x - 27$ .

$81 - 18 \sqrt{x} + x = x - 27$

Move all terms not containing $- 18 \sqrt{x}$ to the right side of the equation.

Subtract $81$ from both sides of the equation.

$- 18 \sqrt{x} + x = x - 27 - 81$

Subtract $x$ from both sides of the equation.

$- 18 \sqrt{x} = x - 27 - 81 - x$

Combine the opposite terms in $x - 27 - 81 - x$ .

Subtract $x$ from $x$ .

$- 18 \sqrt{x} = 0 - 27 - 81$

Subtract $27$ from $0$ .

$- 18 \sqrt{x} = - 27 - 81$

Subtract $81$ from $- 27$ .

$- 18 \sqrt{x} = - 108$

Step 6

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

${(- 18 \sqrt{x})}^{2} = {(- 108)}^{2}$

Step 7

Simplify each side of the equation.

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

${(- 18 x^{\frac{1}{2}})}^{2} = {(- 108)}^{2}$

Simplify the left side.

Simplify ${(- 18 x^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 18 x^{\frac{1}{2}}$ .

${(- 18)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 108)}^{2}$

Raise $- 18$ to the power of $2$ .

$324 {(x^{\frac{1}{2}})}^{2} = {(- 108)}^{2}$

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

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

$324 x^{\frac{1}{2} \cdot 2} = {(- 108)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$324 x^{\frac{1}{2} \cdot 2} = {(- 108)}^{2}$

Rewrite the expression.

$324 x^{1} = {(- 108)}^{2}$

Simplify.

$324 x = {(- 108)}^{2}$

Simplify the right side.

Raise $- 108$ to the power of $2$ .

$324 x = 11664$

Step 8

Divide each term in $324 x = 11664$ by $324$ and simplify.

Divide each term in $324 x = 11664$ by $324$ .

$\frac{324 x}{324} = \frac{11664}{324}$

Simplify the left side.

Cancel the common factor of $324$ .

Cancel the common factor.

$\frac{324 x}{324} = \frac{11664}{324}$

Divide $x$ by $1$ .

$x = \frac{11664}{324}$

Simplify the right side.

Divide $11664$ by $324$ .

$x = 36$

Solve by Factoring square root of x-27=9- square root of x

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