Algebra Examples

The diagram below shows the curve $y = x^{3} - a x^{2}$ ; where a is a positive constant. Find the area of the shaded region in terms of $a$

Step 1

Write the problem as a mathematical expression.

$y = x^{3} - a x^{2}$

Step 2

Rewrite the equation as $x^{3} - a x^{2} = y$ .

$x^{3} - a x^{2} = y$

Step 3

Subtract $x^{3}$ from both sides of the equation.

$- a x^{2} = y - x^{3}$

Step 4

Divide each term in $- a x^{2} = y - x^{3}$ by $- x^{2}$ and simplify.

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Step 4.1

Divide each term in $- a x^{2} = y - x^{3}$ by $- x^{2}$ .

$\frac{- a x^{2}}{- x^{2}} = \frac{y}{- x^{2}} + \frac{- x^{3}}{- x^{2}}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{a x^{2}}{x^{2}} = \frac{y}{- x^{2}} + \frac{- x^{3}}{- x^{2}}$

Step 4.2.2

Cancel the common factor of $x^{2}$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{a x^{2}}{x^{2}} = \frac{y}{- x^{2}} + \frac{- x^{3}}{- x^{2}}$

Step 4.2.2.2

Divide $a$ by $1$ .

$a = \frac{y}{- x^{2}} + \frac{- x^{3}}{- x^{2}}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative in front of the fraction.

$a = - \frac{y}{x^{2}} + \frac{- x^{3}}{- x^{2}}$

Step 4.3.1.2

Dividing two negative values results in a positive value.

$a = - \frac{y}{x^{2}} + \frac{x^{3}}{x^{2}}$

Step 4.3.1.3

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 4.3.1.3.1

Factor $x^{2}$ out of $x^{3}$ .

$a = - \frac{y}{x^{2}} + \frac{x^{2} x}{x^{2}}$

Step 4.3.1.3.2

Cancel the common factors.

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Step 4.3.1.3.2.1

Multiply by $1$ .

$a = - \frac{y}{x^{2}} + \frac{x^{2} x}{x^{2} \cdot 1}$

Step 4.3.1.3.2.2

Cancel the common factor.

$a = - \frac{y}{x^{2}} + \frac{x^{2} x}{x^{2} \cdot 1}$

Step 4.3.1.3.2.3

Rewrite the expression.

$a = - \frac{y}{x^{2}} + \frac{x}{1}$

Step 4.3.1.3.2.4

Divide $x$ by $1$ .

$a = - \frac{y}{x^{2}} + x$