Calculus Examples

$- \frac{24 \sqrt{x}}{x} + 2 = 0$

Step 1

Solve for $24 \sqrt{x}$ .

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Step 1.1

Subtract $2$ from both sides of the equation.

$- \frac{24 \sqrt{x}}{x} = - 2$

Step 1.2

Divide each term in $- \frac{24 \sqrt{x}}{x} = - 2$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- \frac{24 \sqrt{x}}{x} = - 2$ by $- 1$ .

$\frac{- \frac{24 \sqrt{x}}{x}}{- 1} = \frac{- 2}{- 1}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{24 \sqrt{x}}{x}}{1} = \frac{- 2}{- 1}$

Step 1.2.2.2

Divide $\frac{24 \sqrt{x}}{x}$ by $1$ .

$\frac{24 \sqrt{x}}{x} = \frac{- 2}{- 1}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $- 2$ by $- 1$ .

$\frac{24 \sqrt{x}}{x} = 2$

Step 1.3

Multiply both sides by $x$ .

$\frac{24 \sqrt{x}}{x} x = 2 x$

Step 1.4

Simplify the left side.

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Step 1.4.1

Cancel the common factor of $x$ .

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Step 1.4.1.1

Cancel the common factor.

$\frac{24 \sqrt{x}}{x} x = 2 x$

Step 1.4.1.2

Rewrite the expression.

$24 \sqrt{x} = 2 x$

Step 2

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

${(24 \sqrt{x})}^{2} = {(2 x)}^{2}$

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 $x^{\frac{1}{2}}$ .

${(24 x^{\frac{1}{2}})}^{2} = {(2 x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(24 x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $24 x^{\frac{1}{2}}$ .

$24^{2} {(x^{\frac{1}{2}})}^{2} = {(2 x)}^{2}$

Step 3.2.1.2

Raise $24$ to the power of $2$ .

$576 {(x^{\frac{1}{2}})}^{2} = {(2 x)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$576 x^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

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.

$576 x^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$576 x^{1} = {(2 x)}^{2}$

Step 3.2.1.4

Simplify.

$576 x = {(2 x)}^{2}$

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 $2 x$ .

$576 x = 2^{2} x^{2}$

Step 3.3.1.2

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

$576 x = 4 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$576 x - 4 x^{2} = 0$

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 $x$ .

$576 u - 4 u^{2} = 0$

Step 4.2.2

Factor $4 u$ out of $576 u - 4 u^{2}$ .

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Step 4.2.2.1

Factor $4 u$ out of $576 u$ .

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

Step 4.2.2.2

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

$4 u (144) + 4 u (- u) = 0$

Step 4.2.2.3

Factor $4 u$ out of $4 u (144) + 4 u (- u)$ .

$4 u (144 - u) = 0$

Step 4.2.3

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

$4 x (144 - x) = 0$

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 $0$ .

$x = 0$

$144 - x = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $144 - x$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $144 - x$ equal to $0$ .

$144 - x = 0$

Step 4.5.2

Solve $144 - x = 0$ for $x$ .

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Step 4.5.2.1

Subtract $144$ from both sides of the equation.

$- x = - 144$

Step 4.5.2.2

Divide each term in $- x = - 144$ by $- 1$ and simplify.

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Step 4.5.2.2.1

Divide each term in $- x = - 144$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 144}{- 1}$

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.

$\frac{x}{1} = \frac{- 144}{- 1}$

Step 4.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 144}{- 1}$

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Divide $- 144$ by $- 1$ .

$x = 144$

Step 4.6

The final solution is all the values that make $4 x (144 - x) = 0$ true.

$x = 0, 144$

Step 5

Exclude the solutions that do not make $- \frac{24 \sqrt{x}}{x} + 2 = 0$ true.

$x = 144$