Calculus Examples

$y = (7 x^{2} - 6 x + 1) (1 + 2 x)$

Step 1

Set $y$ as a function of $x$ .

$f (x) = (7 x^{2} - 6 x + 1) (1 + 2 x)$

Step 2

Find the derivative.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 7 x^{2} - 6 x + 1$ and $g (x) = 1 + 2 x$ .

$(7 x^{2} - 6 x + 1) \frac{d}{d x} [1 + 2 x] + (1 + 2 x) \frac{d}{d x} [7 x^{2} - 6 x + 1]$

Step 2.2

Differentiate.

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Step 2.2.1

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

$(7 x^{2} - 6 x + 1) (\frac{d}{d x} [1] + \frac{d}{d x} [2 x]) + (1 + 2 x) \frac{d}{d x} [7 x^{2} - 6 x + 1]$

Step 2.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(7 x^{2} - 6 x + 1) (0 + \frac{d}{d x} [2 x]) + (1 + 2 x) \frac{d}{d x} [7 x^{2} - 6 x + 1]$

Step 2.2.3

Add $0$ and $\frac{d}{d x} [2 x]$ .

$(7 x^{2} - 6 x + 1) \frac{d}{d x} [2 x] + (1 + 2 x) \frac{d}{d x} [7 x^{2} - 6 x + 1]$

Step 2.2.4

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

$(7 x^{2} - 6 x + 1) (2 \frac{d}{d x} [x]) + (1 + 2 x) \frac{d}{d x} [7 x^{2} - 6 x + 1]$

Step 2.2.5

Move $2$ to the left of $7 x^{2} - 6 x + 1$ .

$2 \cdot (7 x^{2} - 6 x + 1) \frac{d}{d x} [x] + (1 + 2 x) \frac{d}{d x} [7 x^{2} - 6 x + 1]$

Step 2.2.6

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

$2 (7 x^{2} - 6 x + 1) \cdot 1 + (1 + 2 x) \frac{d}{d x} [7 x^{2} - 6 x + 1]$

Step 2.2.7

Multiply $2$ by $1$ .

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) \frac{d}{d x} [7 x^{2} - 6 x + 1]$

Step 2.2.8

By the Sum Rule, the derivative of $7 x^{2} - 6 x + 1$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1]$ .

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1])$

Step 2.2.9

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1])$

Step 2.2.10

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

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (7 (2 x) + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1])$

Step 2.2.11

Multiply $2$ by $7$ .

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (14 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [1])$

Step 2.2.12

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (14 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 2.2.13

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

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (14 x - 6 \cdot 1 + \frac{d}{d x} [1])$

Step 2.2.14

Multiply $- 6$ by $1$ .

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (14 x - 6 + \frac{d}{d x} [1])$

Step 2.2.15

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (14 x - 6 + 0)$

Step 2.2.16

Add $14 x - 6$ and $0$ .

$2 (7 x^{2} - 6 x + 1) + (1 + 2 x) (14 x - 6)$

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

$2 (7 x^{2}) + 2 (- 6 x) + 2 \cdot 1 + (1 + 2 x) (14 x - 6)$

Step 2.3.2

Apply the distributive property.

$2 (7 x^{2}) + 2 (- 6 x) + 2 \cdot 1 + (1 + 2 x) (14 x) + (1 + 2 x) \cdot - 6$

Step 2.3.3

Apply the distributive property.

$2 (7 x^{2}) + 2 (- 6 x) + 2 \cdot 1 + 1 (14 x) + 2 x (14 x) + (1 + 2 x) \cdot - 6$

Step 2.3.4

Apply the distributive property.

$2 (7 x^{2}) + 2 (- 6 x) + 2 \cdot 1 + 1 (14 x) + 2 x (14 x) + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5

Combine terms.

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Step 2.3.5.1

Multiply $7$ by $2$ .

$14 x^{2} + 2 (- 6 x) + 2 \cdot 1 + 1 (14 x) + 2 x (14 x) + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.2

Multiply $- 6$ by $2$ .

$14 x^{2} - 12 x + 2 \cdot 1 + 1 (14 x) + 2 x (14 x) + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.3

Multiply $2$ by $1$ .

$14 x^{2} - 12 x + 2 + 1 (14 x) + 2 x (14 x) + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.4

Multiply $14$ by $1$ .

$14 x^{2} - 12 x + 2 + 14 x + 2 x (14 x) + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.5

Multiply $14$ by $2$ .

$14 x^{2} - 12 x + 2 + 14 x + 28 x \cdot x + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.6

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

$14 x^{2} - 12 x + 2 + 14 x + 28 (x^{1} x) + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.7

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

$14 x^{2} - 12 x + 2 + 14 x + 28 (x^{1} x^{1}) + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.8

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

$14 x^{2} - 12 x + 2 + 14 x + 28 x^{1 + 1} + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.9

Add $1$ and $1$ .

$14 x^{2} - 12 x + 2 + 14 x + 28 x^{2} + 1 \cdot - 6 + 2 x \cdot - 6$

Step 2.3.5.10

Multiply $- 6$ by $1$ .

$14 x^{2} - 12 x + 2 + 14 x + 28 x^{2} - 6 + 2 x \cdot - 6$

Step 2.3.5.11

Multiply $- 6$ by $2$ .

$14 x^{2} - 12 x + 2 + 14 x + 28 x^{2} - 6 - 12 x$

Step 2.3.5.12

Subtract $12 x$ from $14 x$ .

$14 x^{2} - 12 x + 2 + 2 x + 28 x^{2} - 6$

Step 2.3.5.13

Add $- 12 x$ and $2 x$ .

$14 x^{2} - 10 x + 2 + 28 x^{2} - 6$

Step 2.3.5.14

Add $14 x^{2}$ and $28 x^{2}$ .

$42 x^{2} - 10 x + 2 - 6$

Step 2.3.5.15

Subtract $6$ from $2$ .

$42 x^{2} - 10 x - 4$

Step 3

Set the derivative equal to $0$ then solve the equation $42 x^{2} - 10 x - 4 = 0$ .

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Step 3.1

Factor $2$ out of $42 x^{2} - 10 x - 4$ .

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Step 3.1.1

Factor $2$ out of $42 x^{2}$ .

$2 (21 x^{2}) - 10 x - 4 = 0$

Step 3.1.2

Factor $2$ out of $- 10 x$ .

$2 (21 x^{2}) + 2 (- 5 x) - 4 = 0$

Step 3.1.3

Factor $2$ out of $- 4$ .

$2 (21 x^{2}) + 2 (- 5 x) + 2 (- 2) = 0$

Step 3.1.4

Factor $2$ out of $2 (21 x^{2}) + 2 (- 5 x)$ .

$2 (21 x^{2} - 5 x) + 2 (- 2) = 0$

Step 3.1.5

Factor $2$ out of $2 (21 x^{2} - 5 x) + 2 (- 2)$ .

$2 (21 x^{2} - 5 x - 2) = 0$

Step 3.2

Divide each term in $2 (21 x^{2} - 5 x - 2) = 0$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 (21 x^{2} - 5 x - 2) = 0$ by $2$ .

$\frac{2 (21 x^{2} - 5 x - 2)}{2} = \frac{0}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 (21 x^{2} - 5 x - 2)}{2} = \frac{0}{2}$

Step 3.2.2.1.2

Divide $21 x^{2} - 5 x - 2$ by $1$ .

$21 x^{2} - 5 x - 2 = \frac{0}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $0$ by $2$ .

$21 x^{2} - 5 x - 2 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4

Substitute the values $a = 21$ , $b = - 5$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (21 \cdot - 2)}}{2 \cdot 21}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 21 \cdot - 2}}{2 \cdot 21}$

Step 3.5.1.2

Multiply $- 4 \cdot 21 \cdot - 2$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $21$ .

$x = \frac{5 \pm \sqrt{25 - 84 \cdot - 2}}{2 \cdot 21}$

Step 3.5.1.2.2

Multiply $- 84$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 168}}{2 \cdot 21}$

Step 3.5.1.3

Add $25$ and $168$ .

$x = \frac{5 \pm \sqrt{193}}{2 \cdot 21}$

Step 3.5.2

Multiply $2$ by $21$ .

$x = \frac{5 \pm \sqrt{193}}{42}$

Step 3.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 21 \cdot - 2}}{2 \cdot 21}$

Step 3.6.1.2

Multiply $- 4 \cdot 21 \cdot - 2$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $21$ .

$x = \frac{5 \pm \sqrt{25 - 84 \cdot - 2}}{2 \cdot 21}$

Step 3.6.1.2.2

Multiply $- 84$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 168}}{2 \cdot 21}$

Step 3.6.1.3

Add $25$ and $168$ .

$x = \frac{5 \pm \sqrt{193}}{2 \cdot 21}$

Step 3.6.2

Multiply $2$ by $21$ .

$x = \frac{5 \pm \sqrt{193}}{42}$

Step 3.6.3

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{193}}{42}$

Step 3.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 21 \cdot - 2}}{2 \cdot 21}$

Step 3.7.1.2

Multiply $- 4 \cdot 21 \cdot - 2$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $21$ .

$x = \frac{5 \pm \sqrt{25 - 84 \cdot - 2}}{2 \cdot 21}$

Step 3.7.1.2.2

Multiply $- 84$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 168}}{2 \cdot 21}$

Step 3.7.1.3

Add $25$ and $168$ .

$x = \frac{5 \pm \sqrt{193}}{2 \cdot 21}$

Step 3.7.2

Multiply $2$ by $21$ .

$x = \frac{5 \pm \sqrt{193}}{42}$

Step 3.7.3

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{193}}{42}$

Step 3.8

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{193}}{42}, \frac{5 - \sqrt{193}}{42}$

Step 4

Solve the original function $f (x) = (7 x^{2} - 6 x + 1) (1 + 2 x)$ at $x = \frac{5 + \sqrt{193}}{42}$ .

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Step 4.1

Replace the variable $x$ with $\frac{5 + \sqrt{193}}{42}$ in the expression.

$f (\frac{5 + \sqrt{193}}{42}) = (7 {(\frac{5 + \sqrt{193}}{42})}^{2} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the product rule to $\frac{5 + \sqrt{193}}{42}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (7 (\frac{{(5 + \sqrt{193})}^{2}}{42^{2}}) - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.2

Raise $42$ to the power of $2$ .

$f (\frac{5 + \sqrt{193}}{42}) = (7 (\frac{{(5 + \sqrt{193})}^{2}}{1764}) - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.3

Cancel the common factor of $7$ .

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Step 4.2.1.3.1

Factor $7$ out of $1764$ .

$f (\frac{5 + \sqrt{193}}{42}) = (7 (\frac{{(5 + \sqrt{193})}^{2}}{7 (252)}) - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.3.2

Cancel the common factor.

$f (\frac{5 + \sqrt{193}}{42}) = (7 (\frac{{(5 + \sqrt{193})}^{2}}{7 \cdot 252}) - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.3.3

Rewrite the expression.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{{(5 + \sqrt{193})}^{2}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.4

Rewrite ${(5 + \sqrt{193})}^{2}$ as $(5 + \sqrt{193}) (5 + \sqrt{193})$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{(5 + \sqrt{193}) (5 + \sqrt{193})}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.5

Expand $(5 + \sqrt{193}) (5 + \sqrt{193})$ using the FOIL Method.

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Step 4.2.1.5.1

Apply the distributive property.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{5 (5 + \sqrt{193}) + \sqrt{193} (5 + \sqrt{193})}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.5.2

Apply the distributive property.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{5 \cdot 5 + 5 \sqrt{193} + \sqrt{193} (5 + \sqrt{193})}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.5.3

Apply the distributive property.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{5 \cdot 5 + 5 \sqrt{193} + \sqrt{193} \cdot 5 + \sqrt{193} \sqrt{193}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.6

Simplify and combine like terms.

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Step 4.2.1.6.1

Simplify each term.

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Step 4.2.1.6.1.1

Multiply $5$ by $5$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{25 + 5 \sqrt{193} + \sqrt{193} \cdot 5 + \sqrt{193} \sqrt{193}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.6.1.2

Move $5$ to the left of $\sqrt{193}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{25 + 5 \sqrt{193} + 5 \cdot \sqrt{193} + \sqrt{193} \sqrt{193}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.6.1.3

Combine using the product rule for radicals.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{25 + 5 \sqrt{193} + 5 \sqrt{193} + \sqrt{193 \cdot 193}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.6.1.4

Multiply $193$ by $193$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{25 + 5 \sqrt{193} + 5 \sqrt{193} + \sqrt{37249}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.6.1.5

Rewrite $37249$ as $193^{2}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{25 + 5 \sqrt{193} + 5 \sqrt{193} + \sqrt{193^{2}}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.6.1.6

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{25 + 5 \sqrt{193} + 5 \sqrt{193} + 193}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.6.2

Add $25$ and $193$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{218 + 5 \sqrt{193} + 5 \sqrt{193}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.6.3

Add $5 \sqrt{193}$ and $5 \sqrt{193}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{218 + 10 \sqrt{193}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.7

Cancel the common factor of $218 + 10 \sqrt{193}$ and $252$ .

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Step 4.2.1.7.1

Factor $2$ out of $218$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{2 (109) + 10 \sqrt{193}}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.7.2

Factor $2$ out of $10 \sqrt{193}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{2 (109) + 2 (5 \sqrt{193})}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.7.3

Factor $2$ out of $2 (109) + 2 (5 \sqrt{193})$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{2 (109 + 5 \sqrt{193})}{252} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.7.4

Cancel the common factors.

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Step 4.2.1.7.4.1

Factor $2$ out of $252$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{2 (109 + 5 \sqrt{193})}{2 \cdot 126} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.7.4.2

Cancel the common factor.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{2 (109 + 5 \sqrt{193})}{2 \cdot 126} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.7.4.3

Rewrite the expression.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - 6 \frac{5 + \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.8

Cancel the common factor of $6$ .

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Step 4.2.1.8.1

Factor $6$ out of $- 6$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} + 6 (- 1) (\frac{5 + \sqrt{193}}{42}) + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.8.2

Factor $6$ out of $42$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} + 6 \cdot (- 1 \frac{5 + \sqrt{193}}{6 \cdot 7}) + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.8.3

Cancel the common factor.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} + 6 \cdot (- 1 \frac{5 + \sqrt{193}}{6 \cdot 7}) + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.8.4

Rewrite the expression.

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - 1 \frac{5 + \sqrt{193}}{7} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.1.9

Rewrite $- 1 \frac{5 + \sqrt{193}}{7}$ as $- \frac{5 + \sqrt{193}}{7}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - \frac{5 + \sqrt{193}}{7} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Multiply $\frac{5 + \sqrt{193}}{7}$ by $\frac{18}{18}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - (\frac{5 + \sqrt{193}}{7} \cdot \frac{18}{18}) + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.2.2

Multiply $\frac{5 + \sqrt{193}}{7}$ by $\frac{18}{18}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - \frac{(5 + \sqrt{193}) \cdot 18}{7 \cdot 18} + 1) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.2.3

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

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - \frac{(5 + \sqrt{193}) \cdot 18}{7 \cdot 18} + \frac{1}{1}) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.2.4

Multiply $\frac{1}{1}$ by $\frac{126}{126}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - \frac{(5 + \sqrt{193}) \cdot 18}{7 \cdot 18} + \frac{1}{1} \cdot \frac{126}{126}) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.2.5

Multiply $\frac{1}{1}$ by $\frac{126}{126}$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - \frac{(5 + \sqrt{193}) \cdot 18}{7 \cdot 18} + \frac{126}{126}) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.2.6

Multiply $7$ by $18$ .

$f (\frac{5 + \sqrt{193}}{42}) = (\frac{109 + 5 \sqrt{193}}{126} - \frac{(5 + \sqrt{193}) \cdot 18}{126} + \frac{126}{126}) (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3

Simplify terms.

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Step 4.2.3.1

Combine the numerators over the common denominator.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{109 + 5 \sqrt{193} - (5 + \sqrt{193}) \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.2

Simplify each term.

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Step 4.2.3.2.1

Apply the distributive property.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{109 + 5 \sqrt{193} + (- 1 \cdot 5 - \sqrt{193}) \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.2.2

Multiply $- 1$ by $5$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{109 + 5 \sqrt{193} + (- 5 - \sqrt{193}) \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.2.3

Apply the distributive property.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{109 + 5 \sqrt{193} - 5 \cdot 18 - \sqrt{193} \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.2.4

Multiply $- 5$ by $18$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{109 + 5 \sqrt{193} - 90 - \sqrt{193} \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.2.5

Multiply $18$ by $- 1$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{109 + 5 \sqrt{193} - 90 - 18 \sqrt{193} + 126}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.3

Simplify terms.

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Step 4.2.3.3.1

Subtract $90$ from $109$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{19 + 5 \sqrt{193} - 18 \sqrt{193} + 126}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.3.2

Add $19$ and $126$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 + 5 \sqrt{193} - 18 \sqrt{193}}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.3.3

Subtract $18 \sqrt{193}$ from $5 \sqrt{193}$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 - 13 \sqrt{193}}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{42}))$

Step 4.2.3.3.4

Cancel the common factor of $2$ .

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Step 4.2.3.3.4.1

Factor $2$ out of $42$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 - 13 \sqrt{193}}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{2 (21)}))$

Step 4.2.3.3.4.2

Cancel the common factor.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 - 13 \sqrt{193}}{126} \cdot (1 + 2 (\frac{5 + \sqrt{193}}{2 \cdot 21}))$

Step 4.2.3.3.4.3

Rewrite the expression.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 - 13 \sqrt{193}}{126} \cdot (1 + \frac{5 + \sqrt{193}}{21})$

Step 4.2.3.3.5

Simplify the expression.

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Step 4.2.3.3.5.1

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

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 - 13 \sqrt{193}}{126} \cdot (\frac{21}{21} + \frac{5 + \sqrt{193}}{21})$

Step 4.2.3.3.5.2

Combine the numerators over the common denominator.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 - 13 \sqrt{193}}{126} \cdot \frac{21 + 5 + \sqrt{193}}{21}$

Step 4.2.4

Add $21$ and $5$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 - 13 \sqrt{193}}{126} \cdot \frac{26 + \sqrt{193}}{21}$

Step 4.2.5

Multiply $\frac{145 - 13 \sqrt{193}}{126} \cdot \frac{26 + \sqrt{193}}{21}$ .

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Step 4.2.5.1

Multiply $\frac{145 - 13 \sqrt{193}}{126}$ by $\frac{26 + \sqrt{193}}{21}$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{(145 - 13 \sqrt{193}) (26 + \sqrt{193})}{126 \cdot 21}$

Step 4.2.5.2

Multiply $126$ by $21$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{(145 - 13 \sqrt{193}) (26 + \sqrt{193})}{2646}$

Step 4.2.6

Expand $(145 - 13 \sqrt{193}) (26 + \sqrt{193})$ using the FOIL Method.

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Step 4.2.6.1

Apply the distributive property.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 (26 + \sqrt{193}) - 13 \sqrt{193} (26 + \sqrt{193})}{2646}$

Step 4.2.6.2

Apply the distributive property.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 \cdot 26 + 145 \sqrt{193} - 13 \sqrt{193} (26 + \sqrt{193})}{2646}$

Step 4.2.6.3

Apply the distributive property.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{145 \cdot 26 + 145 \sqrt{193} - 13 \sqrt{193} \cdot 26 - 13 \sqrt{193} \sqrt{193}}{2646}$

Step 4.2.7

Simplify and combine like terms.

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Step 4.2.7.1

Simplify each term.

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Step 4.2.7.1.1

Multiply $145$ by $26$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 13 \sqrt{193} \cdot 26 - 13 \sqrt{193} \sqrt{193}}{2646}$

Step 4.2.7.1.2

Multiply $26$ by $- 13$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 \sqrt{193} \sqrt{193}}{2646}$

Step 4.2.7.1.3

Multiply $- 13 \sqrt{193} \sqrt{193}$ .

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Step 4.2.7.1.3.1

Raise $\sqrt{193}$ to the power of $1$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 (\sqrt{193} \sqrt{193})}{2646}$

Step 4.2.7.1.3.2

Raise $\sqrt{193}$ to the power of $1$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 (\sqrt{193} \sqrt{193})}{2646}$

Step 4.2.7.1.3.3

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

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 {\sqrt{193}}^{1 + 1}}{2646}$

Step 4.2.7.1.3.4

Add $1$ and $1$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 {\sqrt{193}}^{2}}{2646}$

Step 4.2.7.1.4

Rewrite ${\sqrt{193}}^{2}$ as $193$ .

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Step 4.2.7.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{193}$ as $193^{\frac{1}{2}}$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 {(193^{\frac{1}{2}})}^{2}}{2646}$

Step 4.2.7.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 \cdot 193^{\frac{1}{2} \cdot 2}}{2646}$

Step 4.2.7.1.4.3

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

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 \cdot 193^{\frac{2}{2}}}{2646}$

Step 4.2.7.1.4.4

Cancel the common factor of $2$ .

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Step 4.2.7.1.4.4.1

Cancel the common factor.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 \cdot 193^{\frac{2}{2}}}{2646}$

Step 4.2.7.1.4.4.2

Rewrite the expression.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 \cdot 193}{2646}$

Step 4.2.7.1.4.5

Evaluate the exponent.

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 13 \cdot 193}{2646}$

Step 4.2.7.1.5

Multiply $- 13$ by $193$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{3770 + 145 \sqrt{193} - 338 \sqrt{193} - 2509}{2646}$

Step 4.2.7.2

Subtract $2509$ from $3770$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{1261 + 145 \sqrt{193} - 338 \sqrt{193}}{2646}$

Step 4.2.7.3

Subtract $338 \sqrt{193}$ from $145 \sqrt{193}$ .

$f (\frac{5 + \sqrt{193}}{42}) = \frac{1261 - 193 \sqrt{193}}{2646}$

Step 4.2.8

The final answer is $\frac{1261 - 193 \sqrt{193}}{2646}$ .

$\frac{1261 - 193 \sqrt{193}}{2646}$

Step 5

Solve the original function $f (x) = (7 x^{2} - 6 x + 1) (1 + 2 x)$ at $x = \frac{5 - \sqrt{193}}{42}$ .

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Step 5.1

Replace the variable $x$ with $\frac{5 - \sqrt{193}}{42}$ in the expression.

$f (\frac{5 - \sqrt{193}}{42}) = (7 {(\frac{5 - \sqrt{193}}{42})}^{2} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the product rule to $\frac{5 - \sqrt{193}}{42}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (7 (\frac{{(5 - \sqrt{193})}^{2}}{42^{2}}) - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.2

Raise $42$ to the power of $2$ .

$f (\frac{5 - \sqrt{193}}{42}) = (7 (\frac{{(5 - \sqrt{193})}^{2}}{1764}) - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.3

Cancel the common factor of $7$ .

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Step 5.2.1.3.1

Factor $7$ out of $1764$ .

$f (\frac{5 - \sqrt{193}}{42}) = (7 (\frac{{(5 - \sqrt{193})}^{2}}{7 (252)}) - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.3.2

Cancel the common factor.

$f (\frac{5 - \sqrt{193}}{42}) = (7 (\frac{{(5 - \sqrt{193})}^{2}}{7 \cdot 252}) - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.3.3

Rewrite the expression.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{{(5 - \sqrt{193})}^{2}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.4

Rewrite ${(5 - \sqrt{193})}^{2}$ as $(5 - \sqrt{193}) (5 - \sqrt{193})$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{(5 - \sqrt{193}) (5 - \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.5

Expand $(5 - \sqrt{193}) (5 - \sqrt{193})$ using the FOIL Method.

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Step 5.2.1.5.1

Apply the distributive property.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{5 (5 - \sqrt{193}) - \sqrt{193} (5 - \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.5.2

Apply the distributive property.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{5 \cdot 5 + 5 (- \sqrt{193}) - \sqrt{193} (5 - \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.5.3

Apply the distributive property.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{5 \cdot 5 + 5 (- \sqrt{193}) - \sqrt{193} \cdot 5 - \sqrt{193} (- \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6

Simplify and combine like terms.

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Step 5.2.1.6.1

Simplify each term.

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Step 5.2.1.6.1.1

Multiply $5$ by $5$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 + 5 (- \sqrt{193}) - \sqrt{193} \cdot 5 - \sqrt{193} (- \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.2

Multiply $- 1$ by $5$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - \sqrt{193} \cdot 5 - \sqrt{193} (- \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.3

Multiply $5$ by $- 1$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} - \sqrt{193} (- \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.4

Multiply $- \sqrt{193} (- \sqrt{193})$ .

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Step 5.2.1.6.1.4.1

Multiply $- 1$ by $- 1$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + 1 \sqrt{193} \sqrt{193}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.4.2

Multiply $\sqrt{193}$ by $1$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + \sqrt{193} \sqrt{193}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.4.3

Raise $\sqrt{193}$ to the power of $1$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + \sqrt{193} \sqrt{193}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.4.4

Raise $\sqrt{193}$ to the power of $1$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + \sqrt{193} \sqrt{193}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.4.5

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

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + {\sqrt{193}}^{1 + 1}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.4.6

Add $1$ and $1$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + {\sqrt{193}}^{2}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.5

Rewrite ${\sqrt{193}}^{2}$ as $193$ .

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Step 5.2.1.6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{193}$ as $193^{\frac{1}{2}}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + {(193^{\frac{1}{2}})}^{2}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + 193^{\frac{1}{2} \cdot 2}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.5.3

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

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + 193^{\frac{2}{2}}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.1.6.1.5.4.1

Cancel the common factor.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + 193^{\frac{2}{2}}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.5.4.2

Rewrite the expression.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + 193}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.1.5.5

Evaluate the exponent.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{25 - 5 \sqrt{193} - 5 \sqrt{193} + 193}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.2

Add $25$ and $193$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{218 - 5 \sqrt{193} - 5 \sqrt{193}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.6.3

Subtract $5 \sqrt{193}$ from $- 5 \sqrt{193}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{218 - 10 \sqrt{193}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.7

Cancel the common factor of $218 - 10 \sqrt{193}$ and $252$ .

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Step 5.2.1.7.1

Factor $2$ out of $218$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{2 (109) - 10 \sqrt{193}}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.7.2

Factor $2$ out of $- 10 \sqrt{193}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{2 (109) + 2 (- 5 \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.7.3

Factor $2$ out of $2 (109) + 2 (- 5 \sqrt{193})$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{2 (109 - 5 \sqrt{193})}{252} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.7.4

Cancel the common factors.

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Step 5.2.1.7.4.1

Factor $2$ out of $252$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{2 (109 - 5 \sqrt{193})}{2 \cdot 126} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.7.4.2

Cancel the common factor.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{2 (109 - 5 \sqrt{193})}{2 \cdot 126} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.7.4.3

Rewrite the expression.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - 6 \frac{5 - \sqrt{193}}{42} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.8

Cancel the common factor of $6$ .

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Step 5.2.1.8.1

Factor $6$ out of $- 6$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} + 6 (- 1) (\frac{5 - \sqrt{193}}{42}) + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.8.2

Factor $6$ out of $42$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} + 6 \cdot (- 1 \frac{5 - \sqrt{193}}{6 \cdot 7}) + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.8.3

Cancel the common factor.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} + 6 \cdot (- 1 \frac{5 - \sqrt{193}}{6 \cdot 7}) + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.8.4

Rewrite the expression.

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - 1 \frac{5 - \sqrt{193}}{7} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.1.9

Rewrite $- 1 \frac{5 - \sqrt{193}}{7}$ as $- \frac{5 - \sqrt{193}}{7}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - \frac{5 - \sqrt{193}}{7} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.2

Find the common denominator.

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Step 5.2.2.1

Multiply $\frac{5 - \sqrt{193}}{7}$ by $\frac{18}{18}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - (\frac{5 - \sqrt{193}}{7} \cdot \frac{18}{18}) + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.2.2

Multiply $\frac{5 - \sqrt{193}}{7}$ by $\frac{18}{18}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - \frac{(5 - \sqrt{193}) \cdot 18}{7 \cdot 18} + 1) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.2.3

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

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - \frac{(5 - \sqrt{193}) \cdot 18}{7 \cdot 18} + \frac{1}{1}) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.2.4

Multiply $\frac{1}{1}$ by $\frac{126}{126}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - \frac{(5 - \sqrt{193}) \cdot 18}{7 \cdot 18} + \frac{1}{1} \cdot \frac{126}{126}) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.2.5

Multiply $\frac{1}{1}$ by $\frac{126}{126}$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - \frac{(5 - \sqrt{193}) \cdot 18}{7 \cdot 18} + \frac{126}{126}) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.2.6

Multiply $7$ by $18$ .

$f (\frac{5 - \sqrt{193}}{42}) = (\frac{109 - 5 \sqrt{193}}{126} - \frac{(5 - \sqrt{193}) \cdot 18}{126} + \frac{126}{126}) (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3

Simplify terms.

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Step 5.2.3.1

Combine the numerators over the common denominator.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{109 - 5 \sqrt{193} - (5 - \sqrt{193}) \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

Apply the distributive property.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{109 - 5 \sqrt{193} + (- 1 \cdot 5 + \sqrt{193}) \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.2.2

Multiply $- 1$ by $5$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{109 - 5 \sqrt{193} + (- 5 + \sqrt{193}) \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.2.3

Multiply $- - \sqrt{193}$ .

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Step 5.2.3.2.3.1

Multiply $- 1$ by $- 1$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{109 - 5 \sqrt{193} + (- 5 + 1 \sqrt{193}) \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.2.3.2

Multiply $\sqrt{193}$ by $1$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{109 - 5 \sqrt{193} + (- 5 + \sqrt{193}) \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.2.4

Apply the distributive property.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{109 - 5 \sqrt{193} - 5 \cdot 18 + \sqrt{193} \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.2.5

Multiply $- 5$ by $18$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{109 - 5 \sqrt{193} - 90 + \sqrt{193} \cdot 18 + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.2.6

Move $18$ to the left of $\sqrt{193}$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{109 - 5 \sqrt{193} - 90 + 18 \sqrt{193} + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.3

Simplify terms.

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Step 5.2.3.3.1

Subtract $90$ from $109$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{19 - 5 \sqrt{193} + 18 \sqrt{193} + 126}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.3.2

Add $19$ and $126$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 - 5 \sqrt{193} + 18 \sqrt{193}}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.3.3

Add $- 5 \sqrt{193}$ and $18 \sqrt{193}$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 + 13 \sqrt{193}}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{42}))$

Step 5.2.3.3.4

Cancel the common factor of $2$ .

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Step 5.2.3.3.4.1

Factor $2$ out of $42$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 + 13 \sqrt{193}}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{2 (21)}))$

Step 5.2.3.3.4.2

Cancel the common factor.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 + 13 \sqrt{193}}{126} \cdot (1 + 2 (\frac{5 - \sqrt{193}}{2 \cdot 21}))$

Step 5.2.3.3.4.3

Rewrite the expression.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 + 13 \sqrt{193}}{126} \cdot (1 + \frac{5 - \sqrt{193}}{21})$

Step 5.2.3.3.5

Simplify the expression.

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Step 5.2.3.3.5.1

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

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 + 13 \sqrt{193}}{126} \cdot (\frac{21}{21} + \frac{5 - \sqrt{193}}{21})$

Step 5.2.3.3.5.2

Combine the numerators over the common denominator.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 + 13 \sqrt{193}}{126} \cdot \frac{21 + 5 - \sqrt{193}}{21}$

Step 5.2.4

Add $21$ and $5$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 + 13 \sqrt{193}}{126} \cdot \frac{26 - \sqrt{193}}{21}$

Step 5.2.5

Multiply $\frac{145 + 13 \sqrt{193}}{126} \cdot \frac{26 - \sqrt{193}}{21}$ .

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Step 5.2.5.1

Multiply $\frac{145 + 13 \sqrt{193}}{126}$ by $\frac{26 - \sqrt{193}}{21}$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{(145 + 13 \sqrt{193}) (26 - \sqrt{193})}{126 \cdot 21}$

Step 5.2.5.2

Multiply $126$ by $21$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{(145 + 13 \sqrt{193}) (26 - \sqrt{193})}{2646}$

Step 5.2.6

Expand $(145 + 13 \sqrt{193}) (26 - \sqrt{193})$ using the FOIL Method.

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Step 5.2.6.1

Apply the distributive property.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 (26 - \sqrt{193}) + 13 \sqrt{193} (26 - \sqrt{193})}{2646}$

Step 5.2.6.2

Apply the distributive property.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 \cdot 26 + 145 (- \sqrt{193}) + 13 \sqrt{193} (26 - \sqrt{193})}{2646}$

Step 5.2.6.3

Apply the distributive property.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{145 \cdot 26 + 145 (- \sqrt{193}) + 13 \sqrt{193} \cdot 26 + 13 \sqrt{193} (- \sqrt{193})}{2646}$

Step 5.2.7

Simplify and combine like terms.

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Step 5.2.7.1

Simplify each term.

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Step 5.2.7.1.1

Multiply $145$ by $26$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 + 145 (- \sqrt{193}) + 13 \sqrt{193} \cdot 26 + 13 \sqrt{193} (- \sqrt{193})}{2646}$

Step 5.2.7.1.2

Multiply $- 1$ by $145$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 13 \sqrt{193} \cdot 26 + 13 \sqrt{193} (- \sqrt{193})}{2646}$

Step 5.2.7.1.3

Multiply $26$ by $13$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} + 13 \sqrt{193} (- \sqrt{193})}{2646}$

Step 5.2.7.1.4

Multiply $13 \sqrt{193} (- \sqrt{193})$ .

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Step 5.2.7.1.4.1

Multiply $- 1$ by $13$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 \sqrt{193} \sqrt{193}}{2646}$

Step 5.2.7.1.4.2

Raise $\sqrt{193}$ to the power of $1$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 (\sqrt{193} \sqrt{193})}{2646}$

Step 5.2.7.1.4.3

Raise $\sqrt{193}$ to the power of $1$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 (\sqrt{193} \sqrt{193})}{2646}$

Step 5.2.7.1.4.4

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

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 {\sqrt{193}}^{1 + 1}}{2646}$

Step 5.2.7.1.4.5

Add $1$ and $1$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 {\sqrt{193}}^{2}}{2646}$

Step 5.2.7.1.5

Rewrite ${\sqrt{193}}^{2}$ as $193$ .

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Step 5.2.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{193}$ as $193^{\frac{1}{2}}$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 {(193^{\frac{1}{2}})}^{2}}{2646}$

Step 5.2.7.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 \cdot 193^{\frac{1}{2} \cdot 2}}{2646}$

Step 5.2.7.1.5.3

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

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 \cdot 193^{\frac{2}{2}}}{2646}$

Step 5.2.7.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.7.1.5.4.1

Cancel the common factor.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 \cdot 193^{\frac{2}{2}}}{2646}$

Step 5.2.7.1.5.4.2

Rewrite the expression.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 \cdot 193}{2646}$

Step 5.2.7.1.5.5

Evaluate the exponent.

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 13 \cdot 193}{2646}$

Step 5.2.7.1.6

Multiply $- 13$ by $193$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{3770 - 145 \sqrt{193} + 338 \sqrt{193} - 2509}{2646}$

Step 5.2.7.2

Subtract $2509$ from $3770$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{1261 - 145 \sqrt{193} + 338 \sqrt{193}}{2646}$

Step 5.2.7.3

Add $- 145 \sqrt{193}$ and $338 \sqrt{193}$ .

$f (\frac{5 - \sqrt{193}}{42}) = \frac{1261 + 193 \sqrt{193}}{2646}$

Step 5.2.8

The final answer is $\frac{1261 + 193 \sqrt{193}}{2646}$ .

$\frac{1261 + 193 \sqrt{193}}{2646}$

Step 6

The horizontal tangent lines on function $f (x) = (7 x^{2} - 6 x + 1) (1 + 2 x)$ are $y = \frac{1261 - 193 \sqrt{193}}{2646}, y = \frac{1261 + 193 \sqrt{193}}{2646}$ .

$y = \frac{1261 - 193 \sqrt{193}}{2646}, y = \frac{1261 + 193 \sqrt{193}}{2646}$

Step 7