Algebra Examples

$x = 2 \sqrt{6 x - 36}$

Step 1

Subtract $2 \sqrt{6 x - 36}$ from both sides of the equation.

$x - 2 \sqrt{6 x - 36} = 0$

Step 2

Factor $6$ out of $6 x - 36$ .

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Step 2.1

Factor $6$ out of $6 x$ .

$x - 2 \sqrt{6 (x) - 36} = 0$

Step 2.2

Factor $6$ out of $- 36$ .

$x - 2 \sqrt{6 x + 6 \cdot - 6} = 0$

Step 2.3

Factor $6$ out of $6 x + 6 \cdot - 6$ .

$x - 2 \sqrt{6 (x - 6)} = 0$

Step 3

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{6 (x - 6)} = - x$

Step 4

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

${(- 2 \sqrt{6 (x - 6)})}^{2} = {(- x)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 (x - 6)}$ as ${(6 (x - 6))}^{\frac{1}{2}}$ .

${(- 2 {(6 (x - 6))}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(- 2 {(6 (x - 6))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the distributive property.

${(- 2 {(6 x + 6 \cdot - 6)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 5.2.1.2

Simplify the expression.

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Step 5.2.1.2.1

Multiply $6$ by $- 6$ .

${(- 2 {(6 x - 36)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 5.2.1.2.2

Apply the product rule to $- 2 {(6 x - 36)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(6 x - 36)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 5.2.1.2.3

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

$4 {({(6 x - 36)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 5.2.1.2.4

Multiply the exponents in ${({(6 x - 36)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.2.4.1

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

$4 {(6 x - 36)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Step 5.2.1.2.4.2

Cancel the common factor of $2$ .

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Step 5.2.1.2.4.2.1

Cancel the common factor.

$4 {(6 x - 36)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Step 5.2.1.2.4.2.2

Rewrite the expression.

$4 {(6 x - 36)}^{1} = {(- x)}^{2}$

Step 5.2.1.3

Simplify.

$4 (6 x - 36) = {(- x)}^{2}$

Step 5.2.1.4

Apply the distributive property.

$4 (6 x) + 4 \cdot - 36 = {(- x)}^{2}$

Step 5.2.1.5

Multiply.

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Step 5.2.1.5.1

Multiply $6$ by $4$ .

$24 x + 4 \cdot - 36 = {(- x)}^{2}$

Step 5.2.1.5.2

Multiply $4$ by $- 36$ .

$24 x - 144 = {(- x)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

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Step 5.3.1.1

Apply the product rule to $- x$ .

$24 x - 144 = {(- 1)}^{2} x^{2}$

Step 5.3.1.2

Raise $- 1$ to the power of $2$ .

$24 x - 144 = 1 x^{2}$

Step 5.3.1.3

Multiply $x^{2}$ by $1$ .

$24 x - 144 = x^{2}$

Step 6

Solve for $x$ .

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Step 6.1

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

$24 x - 144 - x^{2} = 0$

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Factor $- 1$ out of $24 x - 144 - x^{2}$ .

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Step 6.2.1.1

Reorder the expression.

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Step 6.2.1.1.1

Move $- 144$ .

$24 x - x^{2} - 144 = 0$

Step 6.2.1.1.2

Reorder $24 x$ and $- x^{2}$ .

$- x^{2} + 24 x - 144 = 0$

Step 6.2.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 24 x - 144 = 0$

Step 6.2.1.3

Factor $- 1$ out of $24 x$ .

$- (x^{2}) - (- 24 x) - 144 = 0$

Step 6.2.1.4

Rewrite $- 144$ as $- 1 (144)$ .

$- (x^{2}) - (- 24 x) - 1 \cdot 144 = 0$

Step 6.2.1.5

Factor $- 1$ out of $- (x^{2}) - (- 24 x)$ .

$- (x^{2} - 24 x) - 1 \cdot 144 = 0$

Step 6.2.1.6

Factor $- 1$ out of $- (x^{2} - 24 x) - 1 (144)$ .

$- (x^{2} - 24 x + 144) = 0$

Step 6.2.2

Factor using the perfect square rule.

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Step 6.2.2.1

Rewrite $144$ as $12^{2}$ .

$- (x^{2} - 24 x + 12^{2}) = 0$

Step 6.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$24 x = 2 \cdot x \cdot 12$

Step 6.2.2.3

Rewrite the polynomial.

$- (x^{2} - 2 \cdot x \cdot 12 + 12^{2}) = 0$

Step 6.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 12$ .

$- {(x - 12)}^{2} = 0$

Step 6.3

Divide each term in $- {(x - 12)}^{2} = 0$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- {(x - 12)}^{2} = 0$ by $- 1$ .

$\frac{- {(x - 12)}^{2}}{- 1} = \frac{0}{- 1}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{{(x - 12)}^{2}}{1} = \frac{0}{- 1}$

Step 6.3.2.2

Divide ${(x - 12)}^{2}$ by $1$ .

${(x - 12)}^{2} = \frac{0}{- 1}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $0$ by $- 1$ .

${(x - 12)}^{2} = 0$

Step 6.4

Set the $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 6.5

Add $12$ to both sides of the equation.

$x = 12$