Calculus Examples

$y = x^{2} - 5 x + 4$ , $y = - {(x - 1)}^{2}$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{2} - 5 x + 4 = - {(x - 1)}^{2}$

Step 1.2

Solve $x^{2} - 5 x + 4 = - {(x - 1)}^{2}$ for $x$ .

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Step 1.2.1

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

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Step 1.2.1.1

Rewrite.

$x^{2} - 5 x + 4 = 0 + 0 - {(x - 1)}^{2}$

Step 1.2.1.2

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x^{2} - 5 x + 4 = - ((x - 1) (x - 1))$

Step 1.2.1.3

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.2.1.3.1

Apply the distributive property.

$x^{2} - 5 x + 4 = - (x (x - 1) - 1 (x - 1))$

Step 1.2.1.3.2

Apply the distributive property.

$x^{2} - 5 x + 4 = - (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Step 1.2.1.3.3

Apply the distributive property.

$x^{2} - 5 x + 4 = - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 1.2.1.4

Simplify and combine like terms.

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Step 1.2.1.4.1

Simplify each term.

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Step 1.2.1.4.1.1

Multiply $x$ by $x$ .

$x^{2} - 5 x + 4 = - (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 1.2.1.4.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 5 x + 4 = - (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Step 1.2.1.4.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 5 x + 4 = - (x^{2} - x - 1 x - 1 \cdot - 1)$

Step 1.2.1.4.1.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 5 x + 4 = - (x^{2} - x - x - 1 \cdot - 1)$

Step 1.2.1.4.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - 5 x + 4 = - (x^{2} - x - x + 1)$

Step 1.2.1.4.2

Subtract $x$ from $- x$ .

$x^{2} - 5 x + 4 = - (x^{2} - 2 x + 1)$

Step 1.2.1.5

Apply the distributive property.

$x^{2} - 5 x + 4 = - x^{2} - (- 2 x) - 1 \cdot 1$

Step 1.2.1.6

Simplify.

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Step 1.2.1.6.1

Multiply $- 2$ by $- 1$ .

$x^{2} - 5 x + 4 = - x^{2} + 2 x - 1 \cdot 1$

Step 1.2.1.6.2

Multiply $- 1$ by $1$ .

$x^{2} - 5 x + 4 = - x^{2} + 2 x - 1$

Step 1.2.2

Move all terms containing $x$ to the left side of the equation.

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Step 1.2.2.1

Add $x^{2}$ to both sides of the equation.

$x^{2} - 5 x + 4 + x^{2} = 2 x - 1$

Step 1.2.2.2

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

$x^{2} - 5 x + 4 + x^{2} - 2 x = - 1$

Step 1.2.2.3

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} - 5 x + 4 - 2 x = - 1$

Step 1.2.2.4

Subtract $2 x$ from $- 5 x$ .

$2 x^{2} - 7 x + 4 = - 1$

Step 1.2.3

Add $1$ to both sides of the equation.

$2 x^{2} - 7 x + 4 + 1 = 0$

Step 1.2.4

Add $4$ and $1$ .

$2 x^{2} - 7 x + 5 = 0$

Step 1.2.5

Factor by grouping.

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Step 1.2.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 5 = 10$ and whose sum is $b = - 7$ .

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Step 1.2.5.1.1

Factor $- 7$ out of $- 7 x$ .

$2 x^{2} - 7 x + 5 = 0$

Step 1.2.5.1.2

Rewrite $- 7$ as $- 2$ plus $- 5$

$2 x^{2} + (- 2 - 5) x + 5 = 0$

Step 1.2.5.1.3

Apply the distributive property.

$2 x^{2} - 2 x - 5 x + 5 = 0$

Step 1.2.5.2

Factor out the greatest common factor from each group.

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Step 1.2.5.2.1

Group the first two terms and the last two terms.

$(2 x^{2} - 2 x) - 5 x + 5 = 0$

Step 1.2.5.2.2

Factor out the greatest common factor (GCF) from each group.

$2 x (x - 1) - 5 (x - 1) = 0$

Step 1.2.5.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (2 x - 5) = 0$

Step 1.2.6

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

$x - 1 = 0$

$2 x - 5 = 0 + y = - {(x - 1)}^{2}$

Step 1.2.7

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.7.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.8

Set $2 x - 5$ equal to $0$ and solve for $x$ .

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Step 1.2.8.1

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Step 1.2.8.2

Solve $2 x - 5 = 0$ for $x$ .

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Step 1.2.8.2.1

Add $5$ to both sides of the equation.

$2 x = 5$

Step 1.2.8.2.2

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 1.2.8.2.2.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 1.2.8.2.2.2

Simplify the left side.

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Step 1.2.8.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.8.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 1.2.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 1.2.9

The final solution is all the values that make $(x - 1) (2 x - 5) = 0$ true.

$x = 1, \frac{5}{2}$

Step 1.3

Evaluate $y$ when $x = 1$ .

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Step 1.3.1

Substitute $1$ for $x$ .

$y = - {((1) - 1)}^{2}$

Step 1.3.2

Substitute $1$ for $x$ in $y = - {((1) - 1)}^{2}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = - {(1 - 1)}^{2}$

Step 1.3.2.2

Remove parentheses.

$y = - {((1) - 1)}^{2}$

Step 1.3.2.3

Simplify $- {((1) - 1)}^{2}$ .

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Step 1.3.2.3.1

Subtract $1$ from $1$ .

$y = - 0^{2}$

Step 1.3.2.3.2

Raising $0$ to any positive power yields $0$ .

$y = - 0$

Step 1.3.2.3.3

Multiply $- 1$ by $0$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = \frac{5}{2}$ .

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Step 1.4.1

Substitute $\frac{5}{2}$ for $x$ .

$y = - {((\frac{5}{2}) - 1)}^{2}$

Step 1.4.2

Substitute $\frac{5}{2}$ for $x$ in $y = - {((\frac{5}{2}) - 1)}^{2}$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = - {(\frac{5}{2} - 1)}^{2}$

Step 1.4.2.2

Remove parentheses.

$y = - {((\frac{5}{2}) - 1)}^{2}$

Step 1.4.2.3

Simplify $- {((\frac{5}{2}) - 1)}^{2}$ .

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Step 1.4.2.3.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - {(\frac{5}{2} - 1 \cdot \frac{2}{2})}^{2}$

Step 1.4.2.3.2

Combine $- 1$ and $\frac{2}{2}$ .

$y = - {(\frac{5}{2} + \frac{- 1 \cdot 2}{2})}^{2}$

Step 1.4.2.3.3

Combine the numerators over the common denominator.

$y = - {(\frac{5 - 1 \cdot 2}{2})}^{2}$

Step 1.4.2.3.4

Simplify the numerator.

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Step 1.4.2.3.4.1

Multiply $- 1$ by $2$ .

$y = - {(\frac{5 - 2}{2})}^{2}$

Step 1.4.2.3.4.2

Subtract $2$ from $5$ .

$y = - {(\frac{3}{2})}^{2}$

Step 1.4.2.3.5

Apply the product rule to $\frac{3}{2}$ .

$y = - \frac{3^{2}}{2^{2}}$

Step 1.4.2.3.6

Raise $3$ to the power of $2$ .

$y = - \frac{9}{2^{2}}$

Step 1.4.2.3.7

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

$y = - \frac{9}{4}$

Step 1.5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(1, 0)$

$(\frac{5}{2}, - \frac{9}{4})$

$(1, 0)$

$(\frac{5}{2}, - \frac{9}{4})$

Step 2

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

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Step 2.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$y = - ((x - 1) (x - 1))$

Step 2.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$y = - (x (x - 1) - 1 (x - 1))$

Step 2.2.2

Apply the distributive property.

$y = - (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Step 2.2.3

Apply the distributive property.

$y = - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.3.1.5

Multiply $- 1$ by $- 1$ .

$y = - (x^{2} - x - x + 1)$

Step 2.3.2

Subtract $x$ from $- x$ .

$y = - (x^{2} - 2 x + 1)$

Step 2.4

Apply the distributive property.

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

Step 2.5

Simplify.

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Step 2.5.1

Multiply $- 2$ by $- 1$ .

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

Step 2.5.2

Multiply $- 1$ by $1$ .

$y = - x^{2} + 2 x - 1$

Step 3

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{1}^{\frac{5}{2}} - x^{2} + 2 x - 1 d x - \int_{1}^{\frac{5}{2}} x^{2} - 5 x + 4 d x$

Step 4

Integrate to find the area between $1$ and $\frac{5}{2}$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{1}^{\frac{5}{2}} - x^{2} + 2 x - 1 - (x^{2} - 5 x + 4) d x$

Step 4.2

Simplify each term.

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Step 4.2.1

Apply the distributive property.

$- x^{2} + 2 x - 1 - x^{2} - (- 5 x) - 1 \cdot 4$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $- 5$ by $- 1$ .

$- x^{2} + 2 x - 1 - x^{2} + 5 x - 1 \cdot 4$

Step 4.2.2.2

Multiply $- 1$ by $4$ .

$- x^{2} + 2 x - 1 - x^{2} + 5 x - 4$

$\int_{1}^{\frac{5}{2}} - x^{2} + 2 x - 1 - x^{2} + 5 x - 4 d x$

Step 4.3

Simplify by adding terms.

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Step 4.3.1

Subtract $x^{2}$ from $- x^{2}$ .

$- 2 x^{2} + 2 x - 1 + 5 x - 4$

Step 4.3.2

Add $2 x$ and $5 x$ .

$- 2 x^{2} + 7 x - 1 - 4$

Step 4.3.3

Subtract $4$ from $- 1$ .

$- 2 x^{2} + 7 x - 5$

$\int_{1}^{\frac{5}{2}} - 2 x^{2} + 7 x - 5 d x$

Step 4.4

Split the single integral into multiple integrals.

$\int_{1}^{\frac{5}{2}} - 2 x^{2} d x + \int_{1}^{\frac{5}{2}} 7 x d x + \int_{1}^{\frac{5}{2}} - 5 d x$

Step 4.5

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$- 2 \int_{1}^{\frac{5}{2}} x^{2} d x + \int_{1}^{\frac{5}{2}} 7 x d x + \int_{1}^{\frac{5}{2}} - 5 d x$

Step 4.6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- 2 (\frac{1}{3} x^{3}]_{1}^{\frac{5}{2}}) + \int_{1}^{\frac{5}{2}} 7 x d x + \int_{1}^{\frac{5}{2}} - 5 d x$

Step 4.7

Combine $\frac{1}{3}$ and $x^{3}$ .

$- 2 (\frac{x^{3}}{3}]_{1}^{\frac{5}{2}}) + \int_{1}^{\frac{5}{2}} 7 x d x + \int_{1}^{\frac{5}{2}} - 5 d x$

Step 4.8

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$- 2 (\frac{x^{3}}{3}]_{1}^{\frac{5}{2}}) + 7 \int_{1}^{\frac{5}{2}} x d x + \int_{1}^{\frac{5}{2}} - 5 d x$

Step 4.9

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- 2 (\frac{x^{3}}{3}]_{1}^{\frac{5}{2}}) + 7 (\frac{1}{2} x^{2}]_{1}^{\frac{5}{2}}) + \int_{1}^{\frac{5}{2}} - 5 d x$

Step 4.10

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

$- 2 (\frac{x^{3}}{3}]_{1}^{\frac{5}{2}}) + 7 (\frac{x^{2}}{2}]_{1}^{\frac{5}{2}}) + \int_{1}^{\frac{5}{2}} - 5 d x$

Step 4.11

Apply the constant rule.

$- 2 (\frac{x^{3}}{3}]_{1}^{\frac{5}{2}}) + 7 (\frac{x^{2}}{2}]_{1}^{\frac{5}{2}}) + - 5 x]_{1}^{\frac{5}{2}}$

Step 4.12

Substitute and simplify.

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Step 4.12.1

Evaluate $\frac{x^{3}}{3}$ at $\frac{5}{2}$ and at $1$ .

$- 2 ((\frac{{(\frac{5}{2})}^{3}}{3}) - \frac{1^{3}}{3}) + 7 (\frac{x^{2}}{2}]_{1}^{\frac{5}{2}}) + - 5 x]_{1}^{\frac{5}{2}}$

Step 4.12.2

Evaluate $\frac{x^{2}}{2}$ at $\frac{5}{2}$ and at $1$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1^{3}}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1^{2}}{2}) + - 5 x]_{1}^{\frac{5}{2}}$

Step 4.12.3

Evaluate $- 5 x$ at $\frac{5}{2}$ and at $1$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1^{3}}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1^{2}}{2}) + (- 5 (\frac{5}{2})) + 5 \cdot 1$

Step 4.12.4

Simplify.

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Step 4.12.4.1

One to any power is one.

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1^{2}}{2}) + (- 5 (\frac{5}{2})) + 5 \cdot 1$

Step 4.12.4.2

One to any power is one.

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + (- 5 (\frac{5}{2})) + 5 \cdot 1$

Step 4.12.4.3

Combine $- 5$ and $\frac{5}{2}$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + \frac{- 5 \cdot 5}{2} + 5 \cdot 1$

Step 4.12.4.4

Multiply $- 5$ by $5$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + \frac{- 25}{2} + 5 \cdot 1$

Step 4.12.4.5

Move the negative in front of the fraction.

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) - \frac{25}{2} + 5 \cdot 1$

Step 4.12.4.6

Multiply $5$ by $1$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) - \frac{25}{2} + 5$

Step 4.12.4.7

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) - \frac{25}{2} + 5 \cdot \frac{2}{2}$

Step 4.12.4.8

Combine $5$ and $\frac{2}{2}$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) - \frac{25}{2} + \frac{5 \cdot 2}{2}$

Step 4.12.4.9

Combine the numerators over the common denominator.

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + \frac{- 25 + 5 \cdot 2}{2}$

Step 4.12.4.10

Simplify the numerator.

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Step 4.12.4.10.1

Multiply $5$ by $2$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + \frac{- 25 + 10}{2}$

Step 4.12.4.10.2

Add $- 25$ and $10$ .

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + \frac{- 15}{2}$

Step 4.12.4.11

Move the negative in front of the fraction.

$- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) + 7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) - \frac{15}{2}$

Step 4.12.4.12

To write $- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) - 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) \cdot \frac{2}{2} - \frac{15}{2}$

Step 4.12.4.13

Combine $- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3})$ and $\frac{2}{2}$ .

$7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + \frac{- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) \cdot 2}{2} - \frac{15}{2}$

Step 4.12.4.14

Combine the numerators over the common denominator.

$7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + \frac{- 2 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) \cdot 2 - 15}{2}$

Step 4.12.4.15

Multiply $2$ by $- 2$ .

$7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) + \frac{- 4 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.12.4.16

To write $7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) \cdot \frac{2}{2} + \frac{- 4 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.12.4.17

Combine $7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2})$ and $\frac{2}{2}$ .

$\frac{7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) \cdot 2}{2} + \frac{- 4 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.12.4.18

Combine the numerators over the common denominator.

$\frac{7 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) \cdot 2 - 4 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.12.4.19

Multiply $2$ by $7$ .

$\frac{14 (\frac{{(\frac{5}{2})}^{2}}{2} - \frac{1}{2}) - 4 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13

Simplify.

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Step 4.13.1

Simplify the numerator.

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Step 4.13.1.1

Apply the product rule to $\frac{5}{2}$ .

$\frac{14 (\frac{\frac{5^{2}}{2^{2}}}{2} - \frac{1}{2}) - 4 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.1.2

Raise $5$ to the power of $2$ .

$\frac{14 (\frac{\frac{25}{2^{2}}}{2} - \frac{1}{2}) - 4 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.1.3

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

$\frac{14 (\frac{\frac{25}{4}}{2} - \frac{1}{2}) - 4 (\frac{{(\frac{5}{2})}^{3}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.2

Simplify the numerator.

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Step 4.13.2.1

Apply the product rule to $\frac{5}{2}$ .

$\frac{14 (\frac{\frac{25}{4}}{2} - \frac{1}{2}) - 4 (\frac{\frac{5^{3}}{2^{3}}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.2.2

Raise $5$ to the power of $3$ .

$\frac{14 (\frac{\frac{25}{4}}{2} - \frac{1}{2}) - 4 (\frac{\frac{125}{2^{3}}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.2.3

Raise $2$ to the power of $3$ .

$\frac{14 (\frac{\frac{25}{4}}{2} - \frac{1}{2}) - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3

Simplify the numerator.

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Step 4.13.3.1

Combine the numerators over the common denominator.

$\frac{14 \frac{\frac{25}{4} - 1}{2} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{14 \frac{\frac{25}{4} - 1 \cdot \frac{4}{4}}{2} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.3

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{14 \frac{\frac{25}{4} + \frac{- 1 \cdot 4}{4}}{2} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.4

Combine the numerators over the common denominator.

$\frac{14 \frac{\frac{25 - 1 \cdot 4}{4}}{2} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.5

Simplify the numerator.

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Step 4.13.3.5.1

Multiply $- 1$ by $4$ .

$\frac{14 \frac{\frac{25 - 4}{4}}{2} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.5.2

Subtract $4$ from $25$ .

$\frac{14 \frac{\frac{21}{4}}{2} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.6

Cancel the common factor of $2$ .

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Step 4.13.3.6.1

Factor $2$ out of $14$ .

$\frac{2 (7) \frac{\frac{21}{4}}{2} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.6.2

Cancel the common factor.

$\frac{2 \cdot 7 \frac{\frac{21}{4}}{2} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.6.3

Rewrite the expression.

$\frac{7 (\frac{21}{4}) - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.7

Combine $7$ and $\frac{21}{4}$ .

$\frac{\frac{7 \cdot 21}{4} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.8

Multiply $7$ by $21$ .

$\frac{\frac{147}{4} - 4 (\frac{\frac{125}{8}}{3} - \frac{1}{3}) - 15}{2}$

Step 4.13.3.9

Combine the numerators over the common denominator.

$\frac{\frac{147}{4} - 4 \frac{\frac{125}{8} - 1}{3} - 15}{2}$

Step 4.13.3.10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{\frac{147}{4} - 4 \frac{\frac{125}{8} - 1 \cdot \frac{8}{8}}{3} - 15}{2}$

Step 4.13.3.11

Combine $- 1$ and $\frac{8}{8}$ .

$\frac{\frac{147}{4} - 4 \frac{\frac{125}{8} + \frac{- 1 \cdot 8}{8}}{3} - 15}{2}$

Step 4.13.3.12

Combine the numerators over the common denominator.

$\frac{\frac{147}{4} - 4 \frac{\frac{125 - 1 \cdot 8}{8}}{3} - 15}{2}$

Step 4.13.3.13

Simplify the numerator.

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Step 4.13.3.13.1

Multiply $- 1$ by $8$ .

$\frac{\frac{147}{4} - 4 \frac{\frac{125 - 8}{8}}{3} - 15}{2}$

Step 4.13.3.13.2

Subtract $8$ from $125$ .

$\frac{\frac{147}{4} - 4 \frac{\frac{117}{8}}{3} - 15}{2}$

Step 4.13.3.14

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{147}{4} - 4 (\frac{117}{8} \cdot \frac{1}{3}) - 15}{2}$

Step 4.13.3.15

Cancel the common factor of $3$ .

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Step 4.13.3.15.1

Factor $3$ out of $117$ .

$\frac{\frac{147}{4} - 4 (\frac{3 (39)}{8} \cdot \frac{1}{3}) - 15}{2}$

Step 4.13.3.15.2

Cancel the common factor.

$\frac{\frac{147}{4} - 4 (\frac{3 \cdot 39}{8} \cdot \frac{1}{3}) - 15}{2}$

Step 4.13.3.15.3

Rewrite the expression.

$\frac{\frac{147}{4} - 4 (\frac{39}{8}) - 15}{2}$

Step 4.13.3.16

Cancel the common factor of $4$ .

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Step 4.13.3.16.1

Factor $4$ out of $- 4$ .

$\frac{\frac{147}{4} + 4 (- 1) \frac{39}{8} - 15}{2}$

Step 4.13.3.16.2

Factor $4$ out of $8$ .

$\frac{\frac{147}{4} + 4 \cdot - 1 \frac{39}{4 \cdot 2} - 15}{2}$

Step 4.13.3.16.3

Cancel the common factor.

$\frac{\frac{147}{4} + 4 \cdot - 1 \frac{39}{4 \cdot 2} - 15}{2}$

Step 4.13.3.16.4

Rewrite the expression.

$\frac{\frac{147}{4} - \frac{39}{2} - 15}{2}$

Step 4.13.3.17

To write $- \frac{39}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{147}{4} - \frac{39}{2} \cdot \frac{2}{2} - 15}{2}$

Step 4.13.3.18

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.13.3.18.1

Multiply $\frac{39}{2}$ by $\frac{2}{2}$ .

$\frac{\frac{147}{4} - \frac{39 \cdot 2}{2 \cdot 2} - 15}{2}$

Step 4.13.3.18.2

Multiply $2$ by $2$ .

$\frac{\frac{147}{4} - \frac{39 \cdot 2}{4} - 15}{2}$

Step 4.13.3.19

Combine the numerators over the common denominator.

$\frac{\frac{147 - 39 \cdot 2}{4} - 15}{2}$

Step 4.13.3.20

Simplify the numerator.

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Step 4.13.3.20.1

Multiply $- 39$ by $2$ .

$\frac{\frac{147 - 78}{4} - 15}{2}$

Step 4.13.3.20.2

Subtract $78$ from $147$ .

$\frac{\frac{69}{4} - 15}{2}$

Step 4.13.3.21

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{69}{4} - 15 \cdot \frac{4}{4}}{2}$

Step 4.13.3.22

Combine $- 15$ and $\frac{4}{4}$ .

$\frac{\frac{69}{4} + \frac{- 15 \cdot 4}{4}}{2}$

Step 4.13.3.23

Combine the numerators over the common denominator.

$\frac{\frac{69 - 15 \cdot 4}{4}}{2}$

Step 4.13.3.24

Simplify the numerator.

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Step 4.13.3.24.1

Multiply $- 15$ by $4$ .

$\frac{\frac{69 - 60}{4}}{2}$

Step 4.13.3.24.2

Subtract $60$ from $69$ .

$\frac{\frac{9}{4}}{2}$

Step 4.13.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{9}{4} \cdot \frac{1}{2}$

Step 4.13.5

Multiply $\frac{9}{4} \cdot \frac{1}{2}$ .

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Step 4.13.5.1

Multiply $\frac{9}{4}$ by $\frac{1}{2}$ .

$\frac{9}{4 \cdot 2}$

Step 4.13.5.2

Multiply $4$ by $2$ .

$\frac{9}{8}$

Step 5