Calculus Examples

$2 x - 12 \sqrt{x} = 0$

Step 1

Subtract $2 x$ from both sides of the equation.

$- 12 \sqrt{x} = - 2 x$

Step 2

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

${(- 12 \sqrt{x})}^{2} = {(- 2 x)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- 12 x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- 12 x^{\frac{1}{2}}$ .

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

Step 3.2.1.2

Raise $- 12$ to the power of $2$ .

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

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

Step 3.2.1.3.2.2

Rewrite the expression.

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

Step 3.2.1.4

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Apply the product rule to $- 2 x$ .

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

Step 3.3.1.2

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

$144 x = 4 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $4 x^{2}$ from both sides of the equation.

$144 x - 4 x^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$144 u - 4 u^{2} = 0$

Step 4.2.2

Factor $4 u$ out of $144 u - 4 u^{2}$ .

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Step 4.2.2.1

Factor $4 u$ out of $144 u$ .

$4 u (36) - 4 u^{2} = 0$

Step 4.2.2.2

Factor $4 u$ out of $- 4 u^{2}$ .

$4 u (36) + 4 u (- u) = 0$

Step 4.2.2.3

Factor $4 u$ out of $4 u (36) + 4 u (- u)$ .

$4 u (36 - u) = 0$

Step 4.2.3

Replace all occurrences of $u$ with $x$ .

$4 x (36 - x) = 0$

Step 4.3

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

$x = 0$

$36 - x = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $36 - x$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $36 - x$ equal to $0$ .

$36 - x = 0$

Step 4.5.2

Solve $36 - x = 0$ for $x$ .

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Step 4.5.2.1

Subtract $36$ from both sides of the equation.

$- x = - 36$

Step 4.5.2.2

Divide each term in $- x = - 36$ by $- 1$ and simplify.

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Step 4.5.2.2.1

Divide each term in $- x = - 36$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 36}{- 1}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 36}{- 1}$

Step 4.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 36}{- 1}$

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Divide $- 36$ by $- 1$ .

$x = 36$

Step 4.6

The final solution is all the values that make $4 x (36 - x) = 0$ true.

$x = 0, 36$