Calculus Examples

$f (x) = \frac{1}{4} x^{4} - 3 x^{3} - \frac{81}{2} x^{2} + 729 x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $\frac{1}{4} x^{4} - 3 x^{3} - \frac{81}{2} x^{2} + 729 x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$ .

$f' (x) = \frac{d}{d x} (\frac{1}{4} x^{4}) + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4}]$ .

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Step 1.1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$f' (x) = \frac{1}{4} \cdot \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = \frac{1}{4} \cdot (4 x^{3}) + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.2.3

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

$f' (x) = \frac{4}{4} x^{3} + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$f' (x) = \frac{4 x^{3}}{4} + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.2.5

Cancel the common factor of $4$ .

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Step 1.1.2.5.1

Cancel the common factor.

$f' (x) = \frac{4 x^{3}}{4} + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.2.5.2

Divide $x^{3}$ by $1$ .

$f' (x) = x^{3} + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 3 x^{3}]$ .

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Step 1.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{3}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{3}]$ .

$f' (x) = x^{3} - 3 \frac{d}{d x} x^{3} + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = x^{3} - 3 (3 x^{2}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.3.3

Multiply $3$ by $- 3$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 1.1.4

Evaluate $\frac{d}{d x} [- \frac{81}{2} x^{2}]$ .

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Step 1.1.4.1

Since $- \frac{81}{2}$ is constant with respect to $x$ , the derivative of $- \frac{81}{2} x^{2}$ with respect to $x$ is $- \frac{81}{2} \frac{d}{d x} [x^{2}]$ .

$f' (x) = x^{3} - 9 x^{2} - \frac{81}{2} \cdot \frac{d}{d x} x^{2} + \frac{d}{d x} (729 x)$

Step 1.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = x^{3} - 9 x^{2} - \frac{81}{2} \cdot (2 x) + \frac{d}{d x} (729 x)$

Step 1.1.4.3

Multiply $2$ by $- 1$ .

$f' (x) = x^{3} - 9 x^{2} - 2 (\frac{81}{2}) \cdot x + \frac{d}{d x} (729 x)$

Step 1.1.4.4

Combine $- 2$ and $\frac{81}{2}$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{- 2 \cdot 81}{2} x + \frac{d}{d x} (729 x)$

Step 1.1.4.5

Multiply $- 2$ by $81$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{- 162}{2} x + \frac{d}{d x} (729 x)$

Step 1.1.4.6

Combine $\frac{- 162}{2}$ and $x$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{- 162 x}{2} + \frac{d}{d x} (729 x)$

Step 1.1.4.7

Cancel the common factor of $- 162$ and $2$ .

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Step 1.1.4.7.1

Factor $2$ out of $- 162 x$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2} + \frac{d}{d x} (729 x)$

Step 1.1.4.7.2

Cancel the common factors.

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Step 1.1.4.7.2.1

Factor $2$ out of $2$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2 (1)} + \frac{d}{d x} (729 x)$

Step 1.1.4.7.2.2

Cancel the common factor.

$f' (x) = x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2 \cdot 1} + \frac{d}{d x} (729 x)$

Step 1.1.4.7.2.3

Rewrite the expression.

$f' (x) = x^{3} - 9 x^{2} + \frac{- 81 x}{1} + \frac{d}{d x} (729 x)$

Step 1.1.4.7.2.4

Divide $- 81 x$ by $1$ .

$f' (x) = x^{3} - 9 x^{2} - 81 x + \frac{d}{d x} (729 x)$

Step 1.1.5

Evaluate $\frac{d}{d x} [729 x]$ .

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Step 1.1.5.1

Since $729$ is constant with respect to $x$ , the derivative of $729 x$ with respect to $x$ is $729 \frac{d}{d x} [x]$ .

$f' (x) = x^{3} - 9 x^{2} - 81 x + 729 \frac{d}{d x} (x)$

Step 1.1.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = x^{3} - 9 x^{2} - 81 x + 729 \cdot 1$

Step 1.1.5.3

Multiply $729$ by $1$ .

$f' (x) = x^{3} - 9 x^{2} - 81 x + 729$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{3} - 9 x^{2} - 81 x + 729$ .

$x^{3} - 9 x^{2} - 81 x + 729$

Step 2

Set the first derivative equal to $0$ then solve the equation $x^{3} - 9 x^{2} - 81 x + 729 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$x^{3} - 9 x^{2} - 81 x + 729 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor out the greatest common factor from each group.

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Step 2.2.1.1

Group the first two terms and the last two terms.

$(x^{3} - 9 x^{2}) - 81 x + 729 = 0$

Step 2.2.1.2

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

$x^{2} (x - 9) - 81 (x - 9) = 0$

Step 2.2.2

Factor the polynomial by factoring out the greatest common factor, $x - 9$ .

$(x - 9) (x^{2} - 81) = 0$

Step 2.2.3

Rewrite $81$ as $9^{2}$ .

$(x - 9) (x^{2} - 9^{2}) = 0$

Step 2.2.4

Factor.

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Step 2.2.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$(x - 9) ((x + 9) (x - 9)) = 0$

Step 2.2.4.2

Remove unnecessary parentheses.

$(x - 9) (x + 9) (x - 9) = 0$

Step 2.2.5

Combine exponents.

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Step 2.2.5.1

Raise $x - 9$ to the power of $1$ .

$(x - 9) (x - 9) (x + 9) = 0$

Step 2.2.5.2

Raise $x - 9$ to the power of $1$ .

$(x - 9) (x - 9) (x + 9) = 0$

Step 2.2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(x - 9)}^{1 + 1} (x + 9) = 0$

Step 2.2.5.4

Add $1$ and $1$ .

${(x - 9)}^{2} (x + 9) = 0$

Step 2.3

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

${(x - 9)}^{2} = 0$

$x + 9 = 0$

Step 2.4

Set ${(x - 9)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x - 9)}^{2}$ equal to $0$ .

${(x - 9)}^{2} = 0$

Step 2.4.2

Solve ${(x - 9)}^{2} = 0$ for $x$ .

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Step 2.4.2.1

Set the $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 2.4.2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 2.5

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 2.5.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 2.6

The final solution is all the values that make ${(x - 9)}^{2} (x + 9) = 0$ true.

$x = 9, - 9$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{1}{4} x^{4} - 3 x^{3} - \frac{81}{2} x^{2} + 729 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 9$ .

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Step 4.1.1

Substitute $9$ for $x$ .

$\frac{1}{4} \cdot {(9)}^{4} - 3 {(9)}^{3} - \frac{81}{2} \cdot {(9)}^{2} + 729 (9)$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $9$ to the power of $4$ .

$\frac{1}{4} \cdot 6561 - 3 {(9)}^{3} - \frac{81}{2} \cdot {(9)}^{2} + 729 (9)$

Step 4.1.2.1.2

Combine $\frac{1}{4}$ and $6561$ .

$\frac{6561}{4} - 3 {(9)}^{3} - \frac{81}{2} \cdot {(9)}^{2} + 729 (9)$

Step 4.1.2.1.3

Raise $9$ to the power of $3$ .

$\frac{6561}{4} - 3 \cdot 729 - \frac{81}{2} \cdot {(9)}^{2} + 729 (9)$

Step 4.1.2.1.4

Multiply $- 3$ by $729$ .

$\frac{6561}{4} - 2187 - \frac{81}{2} \cdot {(9)}^{2} + 729 (9)$

Step 4.1.2.1.5

Raise $9$ to the power of $2$ .

$\frac{6561}{4} - 2187 - \frac{81}{2} \cdot 81 + 729 (9)$

Step 4.1.2.1.6

Multiply $- \frac{81}{2} \cdot 81$ .

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Step 4.1.2.1.6.1

Multiply $81$ by $- 1$ .

$\frac{6561}{4} - 2187 - 81 (\frac{81}{2}) + 729 (9)$

Step 4.1.2.1.6.2

Combine $- 81$ and $\frac{81}{2}$ .

$\frac{6561}{4} - 2187 + \frac{- 81 \cdot 81}{2} + 729 (9)$

Step 4.1.2.1.6.3

Multiply $- 81$ by $81$ .

$\frac{6561}{4} - 2187 + \frac{- 6561}{2} + 729 (9)$

Step 4.1.2.1.7

Move the negative in front of the fraction.

$\frac{6561}{4} - 2187 - \frac{6561}{2} + 729 (9)$

Step 4.1.2.1.8

Multiply $729$ by $9$ .

$\frac{6561}{4} - 2187 - \frac{6561}{2} + 6561$

Step 4.1.2.2

Find the common denominator.

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Step 4.1.2.2.1

Write $- 2187$ as a fraction with denominator $1$ .

$\frac{6561}{4} + \frac{- 2187}{1} - \frac{6561}{2} + 6561$

Step 4.1.2.2.2

Multiply $\frac{- 2187}{1}$ by $\frac{4}{4}$ .

$\frac{6561}{4} + \frac{- 2187}{1} \cdot \frac{4}{4} - \frac{6561}{2} + 6561$

Step 4.1.2.2.3

Multiply $\frac{- 2187}{1}$ by $\frac{4}{4}$ .

$\frac{6561}{4} + \frac{- 2187 \cdot 4}{4} - \frac{6561}{2} + 6561$

Step 4.1.2.2.4

Multiply $\frac{6561}{2}$ by $\frac{2}{2}$ .

$\frac{6561}{4} + \frac{- 2187 \cdot 4}{4} - (\frac{6561}{2} \cdot \frac{2}{2}) + 6561$

Step 4.1.2.2.5

Multiply $\frac{6561}{2}$ by $\frac{2}{2}$ .

$\frac{6561}{4} + \frac{- 2187 \cdot 4}{4} - \frac{6561 \cdot 2}{2 \cdot 2} + 6561$

Step 4.1.2.2.6

Write $6561$ as a fraction with denominator $1$ .

$\frac{6561}{4} + \frac{- 2187 \cdot 4}{4} - \frac{6561 \cdot 2}{2 \cdot 2} + \frac{6561}{1}$

Step 4.1.2.2.7

Multiply $\frac{6561}{1}$ by $\frac{4}{4}$ .

$\frac{6561}{4} + \frac{- 2187 \cdot 4}{4} - \frac{6561 \cdot 2}{2 \cdot 2} + \frac{6561}{1} \cdot \frac{4}{4}$

Step 4.1.2.2.8

Multiply $\frac{6561}{1}$ by $\frac{4}{4}$ .

$\frac{6561}{4} + \frac{- 2187 \cdot 4}{4} - \frac{6561 \cdot 2}{2 \cdot 2} + \frac{6561 \cdot 4}{4}$

Step 4.1.2.2.9

Multiply $2$ by $2$ .

$\frac{6561}{4} + \frac{- 2187 \cdot 4}{4} - \frac{6561 \cdot 2}{4} + \frac{6561 \cdot 4}{4}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\frac{6561 - 2187 \cdot 4 - 6561 \cdot 2 + 6561 \cdot 4}{4}$

Step 4.1.2.4

Simplify each term.

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Step 4.1.2.4.1

Multiply $- 2187$ by $4$ .

$\frac{6561 - 8748 - 6561 \cdot 2 + 6561 \cdot 4}{4}$

Step 4.1.2.4.2

Multiply $- 6561$ by $2$ .

$\frac{6561 - 8748 - 13122 + 6561 \cdot 4}{4}$

Step 4.1.2.4.3

Multiply $6561$ by $4$ .

$\frac{6561 - 8748 - 13122 + 26244}{4}$

Step 4.1.2.5

Simplify by adding and subtracting.

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Step 4.1.2.5.1

Subtract $8748$ from $6561$ .

$\frac{- 2187 - 13122 + 26244}{4}$

Step 4.1.2.5.2

Subtract $13122$ from $- 2187$ .

$\frac{- 15309 + 26244}{4}$

Step 4.1.2.5.3

Add $- 15309$ and $26244$ .

$\frac{10935}{4}$

Step 4.2

Evaluate at $x = - 9$ .

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Step 4.2.1

Substitute $- 9$ for $x$ .

$\frac{1}{4} \cdot {(- 9)}^{4} - 3 {(- 9)}^{3} - \frac{81}{2} \cdot {(- 9)}^{2} + 729 (- 9)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Raise $- 9$ to the power of $4$ .

$\frac{1}{4} \cdot 6561 - 3 {(- 9)}^{3} - \frac{81}{2} \cdot {(- 9)}^{2} + 729 (- 9)$

Step 4.2.2.1.2

Combine $\frac{1}{4}$ and $6561$ .

$\frac{6561}{4} - 3 {(- 9)}^{3} - \frac{81}{2} \cdot {(- 9)}^{2} + 729 (- 9)$

Step 4.2.2.1.3

Raise $- 9$ to the power of $3$ .

$\frac{6561}{4} - 3 \cdot - 729 - \frac{81}{2} \cdot {(- 9)}^{2} + 729 (- 9)$

Step 4.2.2.1.4

Multiply $- 3$ by $- 729$ .

$\frac{6561}{4} + 2187 - \frac{81}{2} \cdot {(- 9)}^{2} + 729 (- 9)$

Step 4.2.2.1.5

Raise $- 9$ to the power of $2$ .

$\frac{6561}{4} + 2187 - \frac{81}{2} \cdot 81 + 729 (- 9)$

Step 4.2.2.1.6

Multiply $- \frac{81}{2} \cdot 81$ .

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Step 4.2.2.1.6.1

Multiply $81$ by $- 1$ .

$\frac{6561}{4} + 2187 - 81 (\frac{81}{2}) + 729 (- 9)$

Step 4.2.2.1.6.2

Combine $- 81$ and $\frac{81}{2}$ .

$\frac{6561}{4} + 2187 + \frac{- 81 \cdot 81}{2} + 729 (- 9)$

Step 4.2.2.1.6.3

Multiply $- 81$ by $81$ .

$\frac{6561}{4} + 2187 + \frac{- 6561}{2} + 729 (- 9)$

Step 4.2.2.1.7

Move the negative in front of the fraction.

$\frac{6561}{4} + 2187 - \frac{6561}{2} + 729 (- 9)$

Step 4.2.2.1.8

Multiply $729$ by $- 9$ .

$\frac{6561}{4} + 2187 - \frac{6561}{2} - 6561$

Step 4.2.2.2

Find the common denominator.

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Step 4.2.2.2.1

Write $2187$ as a fraction with denominator $1$ .

$\frac{6561}{4} + \frac{2187}{1} - \frac{6561}{2} - 6561$

Step 4.2.2.2.2

Multiply $\frac{2187}{1}$ by $\frac{4}{4}$ .

$\frac{6561}{4} + \frac{2187}{1} \cdot \frac{4}{4} - \frac{6561}{2} - 6561$

Step 4.2.2.2.3

Multiply $\frac{2187}{1}$ by $\frac{4}{4}$ .

$\frac{6561}{4} + \frac{2187 \cdot 4}{4} - \frac{6561}{2} - 6561$

Step 4.2.2.2.4

Multiply $\frac{6561}{2}$ by $\frac{2}{2}$ .

$\frac{6561}{4} + \frac{2187 \cdot 4}{4} - (\frac{6561}{2} \cdot \frac{2}{2}) - 6561$

Step 4.2.2.2.5

Multiply $\frac{6561}{2}$ by $\frac{2}{2}$ .

$\frac{6561}{4} + \frac{2187 \cdot 4}{4} - \frac{6561 \cdot 2}{2 \cdot 2} - 6561$

Step 4.2.2.2.6

Write $- 6561$ as a fraction with denominator $1$ .

$\frac{6561}{4} + \frac{2187 \cdot 4}{4} - \frac{6561 \cdot 2}{2 \cdot 2} + \frac{- 6561}{1}$

Step 4.2.2.2.7

Multiply $\frac{- 6561}{1}$ by $\frac{4}{4}$ .

$\frac{6561}{4} + \frac{2187 \cdot 4}{4} - \frac{6561 \cdot 2}{2 \cdot 2} + \frac{- 6561}{1} \cdot \frac{4}{4}$

Step 4.2.2.2.8

Multiply $\frac{- 6561}{1}$ by $\frac{4}{4}$ .

$\frac{6561}{4} + \frac{2187 \cdot 4}{4} - \frac{6561 \cdot 2}{2 \cdot 2} + \frac{- 6561 \cdot 4}{4}$

Step 4.2.2.2.9

Multiply $2$ by $2$ .

$\frac{6561}{4} + \frac{2187 \cdot 4}{4} - \frac{6561 \cdot 2}{4} + \frac{- 6561 \cdot 4}{4}$

Step 4.2.2.3

Combine the numerators over the common denominator.

$\frac{6561 + 2187 \cdot 4 - 6561 \cdot 2 - 6561 \cdot 4}{4}$

Step 4.2.2.4

Simplify each term.

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Step 4.2.2.4.1

Multiply $2187$ by $4$ .

$\frac{6561 + 8748 - 6561 \cdot 2 - 6561 \cdot 4}{4}$

Step 4.2.2.4.2

Multiply $- 6561$ by $2$ .

$\frac{6561 + 8748 - 13122 - 6561 \cdot 4}{4}$

Step 4.2.2.4.3

Multiply $- 6561$ by $4$ .

$\frac{6561 + 8748 - 13122 - 26244}{4}$

Step 4.2.2.5

Simplify the expression.

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Step 4.2.2.5.1

Add $6561$ and $8748$ .

$\frac{15309 - 13122 - 26244}{4}$

Step 4.2.2.5.2

Subtract $13122$ from $15309$ .

$\frac{2187 - 26244}{4}$

Step 4.2.2.5.3

Subtract $26244$ from $2187$ .

$\frac{- 24057}{4}$

Step 4.2.2.5.4

Move the negative in front of the fraction.

$- \frac{24057}{4}$

Step 4.3

List all of the points.

$(9, \frac{10935}{4}), (- 9, - \frac{24057}{4})$

Step 5