Calculus Examples

$4 x + 2 y + 6 = 0$ , $(- 5, 4)$

Step 1

The Root Mean Square (RMS) of a function $f$ over a specified interval $[a, b]$ is the square root of the arithmetic mean (average) of the squares of the original values.

$f_{rms} = \sqrt{\frac{1}{b - a} \cdot \int_{a}^{b} f {(y)}^{2} d y}$

Step 2

Substitute the actual values into the formula for the root mean square of a function.

$f_{rms} = \sqrt{\frac{1}{4 + 5} \cdot (\int_{- 5}^{4} {(4 x + 2 y + 6)}^{2} d y)}$

Step 3

Evaluate the integral.

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Step 3.1

Let $u = 4 x + 2 y + 6$ . Then $d u = 2 d y$ , so $\frac{1}{2} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1.1

Let $u = 4 x + 2 y + 6$ . Find $\frac{d u}{d y}$ .

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Step 3.1.1.1

Differentiate $4 x + 2 y + 6$ .

$\frac{d}{d y} [4 x + 2 y + 6]$

Step 3.1.1.2

Differentiate.

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Step 3.1.1.2.1

By the Sum Rule, the derivative of $4 x + 2 y + 6$ with respect to $y$ is $\frac{d}{d y} [4 x] + \frac{d}{d y} [2 y] + \frac{d}{d y} [6]$ .

$\frac{d}{d y} [4 x] + \frac{d}{d y} [2 y] + \frac{d}{d y} [6]$

Step 3.1.1.2.2

Since $4 x$ is constant with respect to $y$ , the derivative of $4 x$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [2 y] + \frac{d}{d y} [6]$

Step 3.1.1.3

Evaluate $\frac{d}{d y} [2 y]$ .

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Step 3.1.1.3.1

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

$0 + 2 \frac{d}{d y} [y] + \frac{d}{d y} [6]$

Step 3.1.1.3.2

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

$0 + 2 \cdot 1 + \frac{d}{d y} [6]$

Step 3.1.1.3.3

Multiply $2$ by $1$ .

$0 + 2 + \frac{d}{d y} [6]$

Step 3.1.1.4

Since $6$ is constant with respect to $y$ , the derivative of $6$ with respect to $y$ is $0$ .

$0 + 2 + 0$

Step 3.1.1.5

Combine terms.

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Step 3.1.1.5.1

Add $0$ and $2$ .

$2 + 0$

Step 3.1.1.5.2

Add $2$ and $0$ .

$2$

Step 3.1.2

Substitute the lower limit in for $y$ in $u = 4 x + 2 y + 6$ .

$u_{lower} = 4 x + 2 \cdot - 5 + 6$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $2$ by $- 5$ .

$u_{lower} = 4 x - 10 + 6$

Step 3.1.3.2

Add $- 10$ and $6$ .

$u_{lower} = 4 x - 4$

Step 3.1.4

Substitute the upper limit in for $y$ in $u = 4 x + 2 y + 6$ .

$u_{upper} = 4 x + 2 \cdot 4 + 6$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Multiply $2$ by $4$ .

$u_{upper} = 4 x + 8 + 6$

Step 3.1.5.2

Add $8$ and $6$ .

$u_{upper} = 4 x + 14$

Step 3.1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 4 x - 4$

$u_{upper} = 4 x + 14$

Step 3.1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{4 x - 4}^{4 x + 14} u^{2} \frac{1}{2} d u$

Step 3.2

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

$\int_{4 x - 4}^{4 x + 14} \frac{u^{2}}{2} d u$

Step 3.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{4 x - 4}^{4 x + 14} u^{2} d u$

Step 3.4

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

$\frac{1}{2} \frac{1}{3} u^{3}]_{4 x - 4}^{4 x + 14}$

Step 3.5

Evaluate $\frac{1}{3} u^{3}$ at $4 x + 14$ and at $4 x - 4$ .

$\frac{1}{2} (\frac{1}{3} {(4 x + 14)}^{3} - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6

Simplify.

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Step 3.6.1

Simplify each term.

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Step 3.6.1.1

Use the Binomial Theorem.

$\frac{1}{2} (\frac{1}{3} ({(4 x)}^{3} + 3 {(4 x)}^{2} \cdot 14 + 3 (4 x) \cdot 14^{2} + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2

Simplify each term.

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Step 3.6.1.2.1

Apply the product rule to $4 x$ .

$\frac{1}{2} (\frac{1}{3} (4^{3} x^{3} + 3 {(4 x)}^{2} \cdot 14 + 3 (4 x) \cdot 14^{2} + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.2

Raise $4$ to the power of $3$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 3 {(4 x)}^{2} \cdot 14 + 3 (4 x) \cdot 14^{2} + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.3

Apply the product rule to $4 x$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 3 (4^{2} x^{2}) \cdot 14 + 3 (4 x) \cdot 14^{2} + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.4

Raise $4$ to the power of $2$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 3 (16 x^{2}) \cdot 14 + 3 (4 x) \cdot 14^{2} + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.5

Multiply $16$ by $3$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 48 x^{2} \cdot 14 + 3 (4 x) \cdot 14^{2} + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.6

Multiply $14$ by $48$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 672 x^{2} + 3 (4 x) \cdot 14^{2} + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.7

Multiply $4$ by $3$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 672 x^{2} + 12 x \cdot 14^{2} + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.8

Raise $14$ to the power of $2$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 672 x^{2} + 12 x \cdot 196 + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.9

Multiply $196$ by $12$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 672 x^{2} + 2352 x + 14^{3}) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.2.10

Raise $14$ to the power of $3$ .

$\frac{1}{2} (\frac{1}{3} (64 x^{3} + 672 x^{2} + 2352 x + 2744) - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.3

Apply the distributive property.

$\frac{1}{2} (\frac{1}{3} (64 x^{3}) + \frac{1}{3} (672 x^{2}) + \frac{1}{3} (2352 x) + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4

Simplify.

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Step 3.6.1.4.1

Multiply $\frac{1}{3} (64 x^{3})$ .

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Step 3.6.1.4.1.1

Combine $64$ and $\frac{1}{3}$ .

$\frac{1}{2} (\frac{64}{3} x^{3} + \frac{1}{3} (672 x^{2}) + \frac{1}{3} (2352 x) + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4.1.2

Combine $\frac{64}{3}$ and $x^{3}$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + \frac{1}{3} (672 x^{2}) + \frac{1}{3} (2352 x) + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4.2

Cancel the common factor of $3$ .

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Step 3.6.1.4.2.1

Factor $3$ out of $672 x^{2}$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + \frac{1}{3} (3 (224 x^{2})) + \frac{1}{3} (2352 x) + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{64 x^{3}}{3} + \frac{1}{3} (3 (224 x^{2})) + \frac{1}{3} (2352 x) + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + \frac{1}{3} (2352 x) + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4.3

Cancel the common factor of $3$ .

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Step 3.6.1.4.3.1

Factor $3$ out of $2352 x$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + \frac{1}{3} (3 (784 x)) + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4.3.2

Cancel the common factor.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + \frac{1}{3} (3 (784 x)) + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4.3.3

Rewrite the expression.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{1}{3} \cdot 2744 - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.4.4

Combine $\frac{1}{3}$ and $2744$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} {(4 x - 4)}^{3})$

Step 3.6.1.5

Use the Binomial Theorem.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} ({(4 x)}^{3} + 3 {(4 x)}^{2} \cdot - 4 + 3 (4 x) {(- 4)}^{2} + {(- 4)}^{3}))$

Step 3.6.1.6

Simplify each term.

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Step 3.6.1.6.1

Apply the product rule to $4 x$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (4^{3} x^{3} + 3 {(4 x)}^{2} \cdot - 4 + 3 (4 x) {(- 4)}^{2} + {(- 4)}^{3}))$

Step 3.6.1.6.2

Raise $4$ to the power of $3$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} + 3 {(4 x)}^{2} \cdot - 4 + 3 (4 x) {(- 4)}^{2} + {(- 4)}^{3}))$

Step 3.6.1.6.3

Apply the product rule to $4 x$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} + 3 (4^{2} x^{2}) \cdot - 4 + 3 (4 x) {(- 4)}^{2} + {(- 4)}^{3}))$

Step 3.6.1.6.4

Raise $4$ to the power of $2$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} + 3 (16 x^{2}) \cdot - 4 + 3 (4 x) {(- 4)}^{2} + {(- 4)}^{3}))$

Step 3.6.1.6.5

Multiply $16$ by $3$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} + 48 x^{2} \cdot - 4 + 3 (4 x) {(- 4)}^{2} + {(- 4)}^{3}))$

Step 3.6.1.6.6

Multiply $- 4$ by $48$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} - 192 x^{2} + 3 (4 x) {(- 4)}^{2} + {(- 4)}^{3}))$

Step 3.6.1.6.7

Multiply $4$ by $3$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} - 192 x^{2} + 12 x {(- 4)}^{2} + {(- 4)}^{3}))$

Step 3.6.1.6.8

Raise $- 4$ to the power of $2$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} - 192 x^{2} + 12 x \cdot 16 + {(- 4)}^{3}))$

Step 3.6.1.6.9

Multiply $16$ by $12$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} - 192 x^{2} + 192 x + {(- 4)}^{3}))$

Step 3.6.1.6.10

Raise $- 4$ to the power of $3$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3} - 192 x^{2} + 192 x - 64))$

Step 3.6.1.7

Apply the distributive property.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{1}{3} (64 x^{3}) - \frac{1}{3} (- 192 x^{2}) - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8

Simplify.

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Step 3.6.1.8.1

Multiply $- \frac{1}{3} (64 x^{3})$ .

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Step 3.6.1.8.1.1

Multiply $64$ by $- 1$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - 64 (\frac{1}{3}) x^{3} - \frac{1}{3} (- 192 x^{2}) - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.1.2

Combine $- 64$ and $\frac{1}{3}$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64}{3} x^{3} - \frac{1}{3} (- 192 x^{2}) - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.1.3

Combine $\frac{- 64}{3}$ and $x^{3}$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} - \frac{1}{3} (- 192 x^{2}) - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.2

Cancel the common factor of $3$ .

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Step 3.6.1.8.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + \frac{- 1}{3} (- 192 x^{2}) - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.2.2

Factor $3$ out of $- 192 x^{2}$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + \frac{- 1}{3} (3 (- 64 x^{2})) - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.2.3

Cancel the common factor.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + \frac{- 1}{3} (3 (- 64 x^{2})) - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.2.4

Rewrite the expression.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} - (- 64 x^{2}) - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.3

Multiply $- 64$ by $- 1$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + 64 x^{2} - \frac{1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.4

Cancel the common factor of $3$ .

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Step 3.6.1.8.4.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + 64 x^{2} + \frac{- 1}{3} (192 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.4.2

Factor $3$ out of $192 x$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + 64 x^{2} + \frac{- 1}{3} (3 (64 x)) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.4.3

Cancel the common factor.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + 64 x^{2} + \frac{- 1}{3} (3 (64 x)) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.4.4

Rewrite the expression.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + 64 x^{2} - (64 x) - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.5

Multiply $64$ by $- 1$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + 64 x^{2} - 64 x - \frac{1}{3} \cdot - 64)$

Step 3.6.1.8.6

Multiply $- \frac{1}{3} \cdot - 64$ .

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Step 3.6.1.8.6.1

Multiply $- 64$ by $- 1$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + 64 x^{2} - 64 x + 64 (\frac{1}{3}))$

Step 3.6.1.8.6.2

Combine $64$ and $\frac{1}{3}$ .

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} + \frac{- 64 x^{3}}{3} + 64 x^{2} - 64 x + \frac{64}{3})$

Step 3.6.1.9

Move the negative in front of the fraction.

$\frac{1}{2} (\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{64 x^{3}}{3} + 64 x^{2} - 64 x + \frac{64}{3})$

Step 3.6.2

Combine the opposite terms in $\frac{64 x^{3}}{3} + 224 x^{2} + 784 x + \frac{2744}{3} - \frac{64 x^{3}}{3} + 64 x^{2} - 64 x + \frac{64}{3}$ .

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Step 3.6.2.1

Subtract $\frac{64 x^{3}}{3}$ from $\frac{64 x^{3}}{3}$ .

$\frac{1}{2} (224 x^{2} + 784 x + \frac{2744}{3} + 0 + 64 x^{2} - 64 x + \frac{64}{3})$

Step 3.6.2.2

Add $224 x^{2} + 784 x + \frac{2744}{3}$ and $0$ .

$\frac{1}{2} (224 x^{2} + 784 x + \frac{2744}{3} + 64 x^{2} - 64 x + \frac{64}{3})$

Step 3.6.3

Combine the numerators over the common denominator.

$\frac{1}{2} (224 x^{2} + 784 x + 64 x^{2} - 64 x + \frac{2744 + 64}{3})$

Step 3.6.4

Add $2744$ and $64$ .

$\frac{1}{2} (224 x^{2} + 784 x + 64 x^{2} - 64 x + \frac{2808}{3})$

Step 3.6.5

Divide $2808$ by $3$ .

$\frac{1}{2} (224 x^{2} + 784 x + 64 x^{2} - 64 x + 936)$

Step 3.6.6

Add $224 x^{2}$ and $64 x^{2}$ .

$\frac{1}{2} (288 x^{2} + 784 x - 64 x + 936)$

Step 3.6.7

Subtract $64 x$ from $784 x$ .

$\frac{1}{2} (288 x^{2} + 720 x + 936)$

Step 3.6.8

Apply the distributive property.

$\frac{1}{2} (288 x^{2}) + \frac{1}{2} (720 x) + \frac{1}{2} \cdot 936$

Step 3.6.9

Simplify.

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Step 3.6.9.1

Cancel the common factor of $2$ .

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Step 3.6.9.1.1

Factor $2$ out of $288 x^{2}$ .

$\frac{1}{2} (2 (144 x^{2})) + \frac{1}{2} (720 x) + \frac{1}{2} \cdot 936$

Step 3.6.9.1.2

Cancel the common factor.

$\frac{1}{2} (2 (144 x^{2})) + \frac{1}{2} (720 x) + \frac{1}{2} \cdot 936$

Step 3.6.9.1.3

Rewrite the expression.

$144 x^{2} + \frac{1}{2} (720 x) + \frac{1}{2} \cdot 936$

Step 3.6.9.2

Cancel the common factor of $2$ .

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Step 3.6.9.2.1

Factor $2$ out of $720 x$ .

$144 x^{2} + \frac{1}{2} (2 (360 x)) + \frac{1}{2} \cdot 936$

Step 3.6.9.2.2

Cancel the common factor.

$144 x^{2} + \frac{1}{2} (2 (360 x)) + \frac{1}{2} \cdot 936$

Step 3.6.9.2.3

Rewrite the expression.

$144 x^{2} + 360 x + \frac{1}{2} \cdot 936$

Step 3.6.9.3

Cancel the common factor of $2$ .

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Step 3.6.9.3.1

Factor $2$ out of $936$ .

$144 x^{2} + 360 x + \frac{1}{2} \cdot (2 (468))$

Step 3.6.9.3.2

Cancel the common factor.

$144 x^{2} + 360 x + \frac{1}{2} \cdot (2 \cdot 468)$

Step 3.6.9.3.3

Rewrite the expression.

$144 x^{2} + 360 x + 468$

Step 4

Simplify the root mean square formula.

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Step 4.1

Add $4$ and $5$ .

$f_{rms} = \sqrt{\frac{1}{9} \cdot (144 x^{2} + 360 x + 468)}$

Step 4.2

Factor $36$ out of $144 x^{2} + 360 x + 468$ .

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Step 4.2.1

Factor $36$ out of $144 x^{2}$ .

$f_{rms} = \sqrt{\frac{1}{9} \cdot (36 (4 x^{2}) + 360 x + 468)}$

Step 4.2.2

Factor $36$ out of $360 x$ .

$f_{rms} = \sqrt{\frac{1}{9} \cdot (36 (4 x^{2}) + 36 (10 x) + 468)}$

Step 4.2.3

Factor $36$ out of $468$ .

$f_{rms} = \sqrt{\frac{1}{9} \cdot (36 (4 x^{2}) + 36 (10 x) + 36 (13))}$

Step 4.2.4

Factor $36$ out of $36 (4 x^{2}) + 36 (10 x)$ .

$f_{rms} = \sqrt{\frac{1}{9} \cdot (36 (4 x^{2} + 10 x) + 36 (13))}$

Step 4.2.5

Factor $36$ out of $36 (4 x^{2} + 10 x) + 36 (13)$ .

$f_{rms} = \sqrt{\frac{1}{9} \cdot (36 (4 x^{2} + 10 x + 13))}$

Step 4.3

Combine $\frac{1}{9}$ and $36$ .

$f_{rms} = \sqrt{\frac{36}{9} \cdot (4 x^{2} + 10 x + 13)}$

Step 4.4

Divide $36$ by $9$ .

$f_{rms} = \sqrt{4 (4 x^{2} + 10 x + 13)}$

Step 4.5

Rewrite $4 (4 x^{2} + 10 x + 13)$ as $2^{2} ({(2 x)}^{2} + 10 x + 13)$ .

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Step 4.5.1

Rewrite $4$ as $2^{2}$ .

$f_{rms} = \sqrt{2^{2} (4 x^{2} + 10 x + 13)}$

Step 4.5.2

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$f_{rms} = \sqrt{2^{2} ({(2 x)}^{2} + 10 x + 13)}$

Step 4.6

Pull terms out from under the radical.

$f_{rms} = 2 \sqrt{{(2 x)}^{2} + 10 x + 13}$

Step 4.7

Simplify the expression.

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Step 4.7.1

Apply the product rule to $2 x$ .

$f_{rms} = 2 \sqrt{2^{2} x^{2} + 10 x + 13}$

Step 4.7.2

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

$f_{rms} = 2 \sqrt{4 x^{2} + 10 x + 13}$

Step 5