Calculus Examples

$60 x - \frac{9}{2} x^{2} = 0$

Step 1

Simplify each term.

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Step 1.1

Combine $x^{2}$ and $\frac{9}{2}$ .

$60 x - \frac{x^{2} \cdot 9}{2} = 0$

Step 1.2

Move $9$ to the left of $x^{2}$ .

$60 x - \frac{9 x^{2}}{2} = 0$

Step 2

Factor $3 x$ out of $60 x - \frac{9 x^{2}}{2}$ .

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Step 2.1

Factor $3 x$ out of $60 x$ .

$3 x (20) - \frac{9 x^{2}}{2} = 0$

Step 2.2

Factor $3 x$ out of $- \frac{9 x^{2}}{2}$ .

$3 x (20) + 3 x (- \frac{3 x}{2}) = 0$

Step 2.3

Factor $3 x$ out of $3 x (20) + 3 x (- \frac{3 x}{2})$ .

$3 x (20 - \frac{3 x}{2}) = 0$

Step 3

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

$x = 0$

$20 - \frac{3 x}{2} = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

Set $20 - \frac{3 x}{2}$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $20 - \frac{3 x}{2}$ equal to $0$ .

$20 - \frac{3 x}{2} = 0$

Step 5.2

Solve $20 - \frac{3 x}{2} = 0$ for $x$ .

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Step 5.2.1

Subtract $20$ from both sides of the equation.

$- \frac{3 x}{2} = - 20$

Step 5.2.2

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} (- \frac{3 x}{2}) = - \frac{2}{3} \cdot - 20$

Step 5.2.3

Simplify both sides of the equation.

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Step 5.2.3.1

Simplify the left side.

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Step 5.2.3.1.1

Simplify $- \frac{2}{3} (- \frac{3 x}{2})$ .

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Step 5.2.3.1.1.1

Cancel the common factor of $2$ .

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Step 5.2.3.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} (- \frac{3 x}{2}) = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.1.2

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x) = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.1.1.2.1

Factor $3$ out of $- 3 x$ .

$\frac{- 1}{3} (3 (- x)) = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.2.3

Rewrite the expression.

$- - x = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.3

Multiply.

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Step 5.2.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{2}{3} \cdot - 20$

Step 5.2.3.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{2}{3} \cdot - 20$

Step 5.2.3.2

Simplify the right side.

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Step 5.2.3.2.1

Multiply $- \frac{2}{3} \cdot - 20$ .

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Step 5.2.3.2.1.1

Multiply $- 20$ by $- 1$ .

$x = 20 (\frac{2}{3})$

Step 5.2.3.2.1.2

Combine $20$ and $\frac{2}{3}$ .

$x = \frac{20 \cdot 2}{3}$

Step 5.2.3.2.1.3

Multiply $20$ by $2$ .

$x = \frac{40}{3}$

Step 6

The final solution is all the values that make $3 x (20 - \frac{3 x}{2}) = 0$ true.

$x = 0, \frac{40}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{40}{3}$

Decimal Form:

$x = 0, 13.\overline{3}$

Mixed Number Form:

$x = 0, 13 \frac{1}{3}$