Finite Math Examples

$p = q^{2} + 8 q + 16$ , $p = 118 q^{2} + 424$

Step 1

Eliminate the equal sides of each equation and combine.

$q^{2} + 8 q + 16 = 118 q^{2} + 424$

Step 2

Solve $q^{2} + 8 q + 16 = 118 q^{2} + 424$ for $q$ .

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Step 2.1

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

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Step 2.1.1

Subtract $118 q^{2}$ from both sides of the equation.

$q^{2} + 8 q + 16 - 118 q^{2} = 424$

Step 2.1.2

Subtract $118 q^{2}$ from $q^{2}$ .

$- 117 q^{2} + 8 q + 16 = 424$

Step 2.2

Move all terms to the left side of the equation and simplify.

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Step 2.2.1

Subtract $424$ from both sides of the equation.

$- 117 q^{2} + 8 q + 16 - 424 = 0$

Step 2.2.2

Subtract $424$ from $16$ .

$- 117 q^{2} + 8 q - 408 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + p = 118 q^{2} + 424$

Step 2.4

Substitute the values $a = - 117$ , $b = 8$ , and $c = - 408$ into the quadratic formula and solve for $q$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (- 117 \cdot - 408)}}{2 \cdot - 117} + p = 118 q^{2} + 424$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $8$ to the power of $2$ .

$q = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 117 \cdot - 408}}{2 \cdot - 117}$

Step 2.5.1.2

Multiply $- 4 \cdot - 117 \cdot - 408$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $- 117$ .

$q = \frac{- 8 \pm \sqrt{64 + 468 \cdot - 408}}{2 \cdot - 117}$

Step 2.5.1.2.2

Multiply $468$ by $- 408$ .

$q = \frac{- 8 \pm \sqrt{64 - 190944}}{2 \cdot - 117}$

Step 2.5.1.3

Subtract $190944$ from $64$ .

$q = \frac{- 8 \pm \sqrt{- 190880}}{2 \cdot - 117}$

Step 2.5.1.4

Rewrite $- 190880$ as $- 1 (190880)$ .

$q = \frac{- 8 \pm \sqrt{- 1 \cdot 190880}}{2 \cdot - 117}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (190880)}$ as $\sqrt{- 1} \cdot \sqrt{190880}$ .

$q = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{190880}}{2 \cdot - 117}$

Step 2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$q = \frac{- 8 \pm i \cdot \sqrt{190880}}{2 \cdot - 117}$

Step 2.5.1.7

Rewrite $190880$ as $4^{2} \cdot 11930$ .

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Step 2.5.1.7.1

Factor $16$ out of $190880$ .

$q = \frac{- 8 \pm i \cdot \sqrt{16 (11930)}}{2 \cdot - 117}$

Step 2.5.1.7.2

Rewrite $16$ as $4^{2}$ .

$q = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 11930}}{2 \cdot - 117}$

Step 2.5.1.8

Pull terms out from under the radical.

$q = \frac{- 8 \pm i \cdot (4 \sqrt{11930})}{2 \cdot - 117}$

Step 2.5.1.9

Move $4$ to the left of $i$ .

$q = \frac{- 8 \pm 4 i \sqrt{11930}}{2 \cdot - 117}$

Step 2.5.2

Multiply $2$ by $- 117$ .

$q = \frac{- 8 \pm 4 i \sqrt{11930}}{- 234}$

Step 2.5.3

Simplify $\frac{- 8 \pm 4 i \sqrt{11930}}{- 234}$ .

$q = \frac{4 \pm 2 i \sqrt{11930}}{117}$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $8$ to the power of $2$ .

$q = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 117 \cdot - 408}}{2 \cdot - 117}$

Step 2.6.1.2

Multiply $- 4 \cdot - 117 \cdot - 408$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $- 117$ .

$q = \frac{- 8 \pm \sqrt{64 + 468 \cdot - 408}}{2 \cdot - 117}$

Step 2.6.1.2.2

Multiply $468$ by $- 408$ .

$q = \frac{- 8 \pm \sqrt{64 - 190944}}{2 \cdot - 117}$

Step 2.6.1.3

Subtract $190944$ from $64$ .

$q = \frac{- 8 \pm \sqrt{- 190880}}{2 \cdot - 117}$

Step 2.6.1.4

Rewrite $- 190880$ as $- 1 (190880)$ .

$q = \frac{- 8 \pm \sqrt{- 1 \cdot 190880}}{2 \cdot - 117}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (190880)}$ as $\sqrt{- 1} \cdot \sqrt{190880}$ .

$q = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{190880}}{2 \cdot - 117}$

Step 2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$q = \frac{- 8 \pm i \cdot \sqrt{190880}}{2 \cdot - 117}$

Step 2.6.1.7

Rewrite $190880$ as $4^{2} \cdot 11930$ .

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Step 2.6.1.7.1

Factor $16$ out of $190880$ .

$q = \frac{- 8 \pm i \cdot \sqrt{16 (11930)}}{2 \cdot - 117}$

Step 2.6.1.7.2

Rewrite $16$ as $4^{2}$ .

$q = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 11930}}{2 \cdot - 117}$

Step 2.6.1.8

Pull terms out from under the radical.

$q = \frac{- 8 \pm i \cdot (4 \sqrt{11930})}{2 \cdot - 117}$

Step 2.6.1.9

Move $4$ to the left of $i$ .

$q = \frac{- 8 \pm 4 i \sqrt{11930}}{2 \cdot - 117}$

Step 2.6.2

Multiply $2$ by $- 117$ .

$q = \frac{- 8 \pm 4 i \sqrt{11930}}{- 234}$

Step 2.6.3

Simplify $\frac{- 8 \pm 4 i \sqrt{11930}}{- 234}$ .

$q = \frac{4 \pm 2 i \sqrt{11930}}{117}$

Step 2.6.4

Change the $\pm$ to $+$ .

$q = \frac{4 + 2 i \sqrt{11930}}{117}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $8$ to the power of $2$ .

$q = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 117 \cdot - 408}}{2 \cdot - 117}$

Step 2.7.1.2

Multiply $- 4 \cdot - 117 \cdot - 408$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $- 117$ .

$q = \frac{- 8 \pm \sqrt{64 + 468 \cdot - 408}}{2 \cdot - 117}$

Step 2.7.1.2.2

Multiply $468$ by $- 408$ .

$q = \frac{- 8 \pm \sqrt{64 - 190944}}{2 \cdot - 117}$

Step 2.7.1.3

Subtract $190944$ from $64$ .

$q = \frac{- 8 \pm \sqrt{- 190880}}{2 \cdot - 117}$

Step 2.7.1.4

Rewrite $- 190880$ as $- 1 (190880)$ .

$q = \frac{- 8 \pm \sqrt{- 1 \cdot 190880}}{2 \cdot - 117}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (190880)}$ as $\sqrt{- 1} \cdot \sqrt{190880}$ .

$q = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{190880}}{2 \cdot - 117}$

Step 2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$q = \frac{- 8 \pm i \cdot \sqrt{190880}}{2 \cdot - 117}$

Step 2.7.1.7

Rewrite $190880$ as $4^{2} \cdot 11930$ .

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Step 2.7.1.7.1

Factor $16$ out of $190880$ .

$q = \frac{- 8 \pm i \cdot \sqrt{16 (11930)}}{2 \cdot - 117}$

Step 2.7.1.7.2

Rewrite $16$ as $4^{2}$ .

$q = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 11930}}{2 \cdot - 117}$

Step 2.7.1.8

Pull terms out from under the radical.

$q = \frac{- 8 \pm i \cdot (4 \sqrt{11930})}{2 \cdot - 117}$

Step 2.7.1.9

Move $4$ to the left of $i$ .

$q = \frac{- 8 \pm 4 i \sqrt{11930}}{2 \cdot - 117}$

Step 2.7.2

Multiply $2$ by $- 117$ .

$q = \frac{- 8 \pm 4 i \sqrt{11930}}{- 234}$

Step 2.7.3

Simplify $\frac{- 8 \pm 4 i \sqrt{11930}}{- 234}$ .

$q = \frac{4 \pm 2 i \sqrt{11930}}{117}$

Step 2.7.4

Change the $\pm$ to $-$ .

$q = \frac{4 - 2 i \sqrt{11930}}{117}$

Step 2.8

The final answer is the combination of both solutions.

$q = \frac{4 + 2 i \sqrt{11930}}{117}, \frac{4 - 2 i \sqrt{11930}}{117}$

Step 3

Evaluate $p$ when $q = \frac{4 + 2 i \sqrt{11930}}{117}$ .

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Step 3.1

Substitute $\frac{4 + 2 i \sqrt{11930}}{117}$ for $q$ .

$p = 118 {(\frac{4 + 2 i \sqrt{11930}}{117})}^{2} + 424$

Step 3.2

Simplify $118 {(\frac{4 + 2 i \sqrt{11930}}{117})}^{2} + 424$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the product rule to $\frac{4 + 2 i \sqrt{11930}}{117}$ .

$p = 118 \frac{{(4 + 2 i \sqrt{11930})}^{2}}{117^{2}} + 424$

Step 3.2.1.2

Raise $117$ to the power of $2$ .

$p = 118 \frac{{(4 + 2 i \sqrt{11930})}^{2}}{13689} + 424$

Step 3.2.1.3

Rewrite ${(4 + 2 i \sqrt{11930})}^{2}$ as $(4 + 2 i \sqrt{11930}) (4 + 2 i \sqrt{11930})$ .

$p = 118 \frac{(4 + 2 i \sqrt{11930}) (4 + 2 i \sqrt{11930})}{13689} + 424$

Step 3.2.1.4

Expand $(4 + 2 i \sqrt{11930}) (4 + 2 i \sqrt{11930})$ using the FOIL Method.

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Step 3.2.1.4.1

Apply the distributive property.

$p = 118 \frac{4 (4 + 2 i \sqrt{11930}) + 2 i \sqrt{11930} (4 + 2 i \sqrt{11930})}{13689} + 424$

Step 3.2.1.4.2

Apply the distributive property.

$p = 118 \frac{4 \cdot 4 + 4 (2 i \sqrt{11930}) + 2 i \sqrt{11930} (4 + 2 i \sqrt{11930})}{13689} + 424$

Step 3.2.1.4.3

Apply the distributive property.

$p = 118 \frac{4 \cdot 4 + 4 (2 i \sqrt{11930}) + 2 i \sqrt{11930} \cdot 4 + 2 i \sqrt{11930} (2 i \sqrt{11930})}{13689} + 424$

Step 3.2.1.5

Simplify and combine like terms.

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Step 3.2.1.5.1

Simplify each term.

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Step 3.2.1.5.1.1

Multiply $4$ by $4$ .

$p = 118 \frac{16 + 4 (2 i \sqrt{11930}) + 2 i \sqrt{11930} \cdot 4 + 2 i \sqrt{11930} (2 i \sqrt{11930})}{13689} + 424$

Step 3.2.1.5.1.2

Multiply $2$ by $4$ .

$p = 118 \frac{16 + 8 (i \sqrt{11930}) + 2 i \sqrt{11930} \cdot 4 + 2 i \sqrt{11930} (2 i \sqrt{11930})}{13689} + 424$

Step 3.2.1.5.1.3

Multiply $4$ by $2$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 2 i \sqrt{11930} (2 i \sqrt{11930})}{13689} + 424$

Step 3.2.1.5.1.4

Multiply $2 i \sqrt{11930} (2 i \sqrt{11930})$ .

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Step 3.2.1.5.1.4.1

Multiply $2$ by $2$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 i \sqrt{11930} (i \sqrt{11930})}{13689} + 424$

Step 3.2.1.5.1.4.2

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

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 (i^{1} i) \sqrt{11930} \sqrt{11930}}{13689} + 424$

Step 3.2.1.5.1.4.3

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

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 (i^{1} i^{1}) \sqrt{11930} \sqrt{11930}}{13689} + 424$

Step 3.2.1.5.1.4.4

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

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 i^{1 + 1} \sqrt{11930} \sqrt{11930}}{13689} + 424$

Step 3.2.1.5.1.4.5

Add $1$ and $1$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 i^{2} \sqrt{11930} \sqrt{11930}}{13689} + 424$

Step 3.2.1.5.1.4.6

Raise $\sqrt{11930}$ to the power of $1$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 i^{2} ({\sqrt{11930}}^{1} \sqrt{11930})}{13689} + 424$

Step 3.2.1.5.1.4.7

Raise $\sqrt{11930}$ to the power of $1$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 i^{2} ({\sqrt{11930}}^{1} {\sqrt{11930}}^{1})}{13689} + 424$

Step 3.2.1.5.1.4.8

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

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 i^{2} {\sqrt{11930}}^{1 + 1}}{13689} + 424$

Step 3.2.1.5.1.4.9

Add $1$ and $1$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 i^{2} {\sqrt{11930}}^{2}}{13689} + 424$

Step 3.2.1.5.1.5

Rewrite $i^{2}$ as $- 1$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} + 4 \cdot - 1 {\sqrt{11930}}^{2}}{13689} + 424$

Step 3.2.1.5.1.6

Multiply $4$ by $- 1$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} - 4 {\sqrt{11930}}^{2}}{13689} + 424$

Step 3.2.1.5.1.7

Rewrite ${\sqrt{11930}}^{2}$ as $11930$ .

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Step 3.2.1.5.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11930}$ as $11930^{\frac{1}{2}}$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} - 4 {(11930^{\frac{1}{2}})}^{2}}{13689} + 424$

Step 3.2.1.5.1.7.2

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

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} - 4 \cdot 11930^{\frac{1}{2} \cdot 2}}{13689} + 424$

Step 3.2.1.5.1.7.3

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

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} - 4 \cdot 11930^{\frac{2}{2}}}{13689} + 424$

Step 3.2.1.5.1.7.4

Cancel the common factor of $2$ .

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Step 3.2.1.5.1.7.4.1

Cancel the common factor.

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} - 4 \cdot 11930^{\frac{2}{2}}}{13689} + 424$

Step 3.2.1.5.1.7.4.2

Rewrite the expression.

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} - 4 \cdot 11930^{1}}{13689} + 424$

Step 3.2.1.5.1.7.5

Evaluate the exponent.

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} - 4 \cdot 11930}{13689} + 424$

Step 3.2.1.5.1.8

Multiply $- 4$ by $11930$ .

$p = 118 \frac{16 + 8 i \sqrt{11930} + 8 i \sqrt{11930} - 47720}{13689} + 424$

Step 3.2.1.5.2

Subtract $47720$ from $16$ .

$p = 118 \frac{8 i \sqrt{11930} + 8 i \sqrt{11930} - 47704}{13689} + 424$

Step 3.2.1.5.3

Add $8 i \sqrt{11930}$ and $8 i \sqrt{11930}$ .

$p = 118 \frac{16 i \sqrt{11930} - 47704}{13689} + 424$

Step 3.2.1.6

Reorder $16 i \sqrt{11930}$ and $- 47704$ .

$p = 118 \frac{- 47704 + 16 i \sqrt{11930}}{13689} + 424$

Step 3.2.1.7

Combine $118$ and $\frac{- 47704 + 16 i \sqrt{11930}}{13689}$ .

$p = \frac{118 (- 47704 + 16 i \sqrt{11930})}{13689} + 424$

Step 3.2.2

To write $424$ as a fraction with a common denominator, multiply by $\frac{13689}{13689}$ .

$p = \frac{118 (- 47704 + 16 i \sqrt{11930})}{13689} + 424 \cdot \frac{13689}{13689}$

Step 3.2.3

Combine $424$ and $\frac{13689}{13689}$ .

$p = \frac{118 (- 47704 + 16 i \sqrt{11930})}{13689} + \frac{424 \cdot 13689}{13689}$

Step 3.2.4

Combine the numerators over the common denominator.

$p = \frac{175064 + 1888 i \sqrt{11930}}{13689}$

Step 4

Evaluate $p$ when $q = \frac{4 - 2 i \sqrt{11930}}{117}$ .

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Step 4.1

Substitute $\frac{4 - 2 i \sqrt{11930}}{117}$ for $q$ .

$p = 118 {(\frac{4 - 2 i \sqrt{11930}}{117})}^{2} + 424$

Step 4.2

Simplify $118 {(\frac{4 - 2 i \sqrt{11930}}{117})}^{2} + 424$ .

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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 $\frac{4 - 2 i \sqrt{11930}}{117}$ .

$p = 118 \frac{{(4 - 2 i \sqrt{11930})}^{2}}{117^{2}} + 424$

Step 4.2.1.2

Raise $117$ to the power of $2$ .

$p = 118 \frac{{(4 - 2 i \sqrt{11930})}^{2}}{13689} + 424$

Step 4.2.1.3

Rewrite ${(4 - 2 i \sqrt{11930})}^{2}$ as $(4 - 2 i \sqrt{11930}) (4 - 2 i \sqrt{11930})$ .

$p = 118 \frac{(4 - 2 i \sqrt{11930}) (4 - 2 i \sqrt{11930})}{13689} + 424$

Step 4.2.1.4

Expand $(4 - 2 i \sqrt{11930}) (4 - 2 i \sqrt{11930})$ using the FOIL Method.

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Step 4.2.1.4.1

Apply the distributive property.

$p = 118 \frac{4 (4 - 2 i \sqrt{11930}) - 2 i \sqrt{11930} (4 - 2 i \sqrt{11930})}{13689} + 424$

Step 4.2.1.4.2

Apply the distributive property.

$p = 118 \frac{4 \cdot 4 + 4 (- 2 i \sqrt{11930}) - 2 i \sqrt{11930} (4 - 2 i \sqrt{11930})}{13689} + 424$

Step 4.2.1.4.3

Apply the distributive property.

$p = 118 \frac{4 \cdot 4 + 4 (- 2 i \sqrt{11930}) - 2 i \sqrt{11930} \cdot 4 - 2 i \sqrt{11930} (- 2 i \sqrt{11930})}{13689} + 424$

Step 4.2.1.5

Simplify and combine like terms.

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Step 4.2.1.5.1

Simplify each term.

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Step 4.2.1.5.1.1

Multiply $4$ by $4$ .

$p = 118 \frac{16 + 4 (- 2 i \sqrt{11930}) - 2 i \sqrt{11930} \cdot 4 - 2 i \sqrt{11930} (- 2 i \sqrt{11930})}{13689} + 424$

Step 4.2.1.5.1.2

Multiply $- 2$ by $4$ .

$p = 118 \frac{16 - 8 (i \sqrt{11930}) - 2 i \sqrt{11930} \cdot 4 - 2 i \sqrt{11930} (- 2 i \sqrt{11930})}{13689} + 424$

Step 4.2.1.5.1.3

Multiply $4$ by $- 2$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 2 i \sqrt{11930} (- 2 i \sqrt{11930})}{13689} + 424$

Step 4.2.1.5.1.4

Multiply $- 2 i \sqrt{11930} (- 2 i \sqrt{11930})$ .

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Step 4.2.1.5.1.4.1

Multiply $- 2$ by $- 2$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 i \sqrt{11930} (i \sqrt{11930})}{13689} + 424$

Step 4.2.1.5.1.4.2

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

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 (i^{1} i) \sqrt{11930} \sqrt{11930}}{13689} + 424$

Step 4.2.1.5.1.4.3

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

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 (i^{1} i^{1}) \sqrt{11930} \sqrt{11930}}{13689} + 424$

Step 4.2.1.5.1.4.4

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

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 i^{1 + 1} \sqrt{11930} \sqrt{11930}}{13689} + 424$

Step 4.2.1.5.1.4.5

Add $1$ and $1$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 i^{2} \sqrt{11930} \sqrt{11930}}{13689} + 424$

Step 4.2.1.5.1.4.6

Raise $\sqrt{11930}$ to the power of $1$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 i^{2} ({\sqrt{11930}}^{1} \sqrt{11930})}{13689} + 424$

Step 4.2.1.5.1.4.7

Raise $\sqrt{11930}$ to the power of $1$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 i^{2} ({\sqrt{11930}}^{1} {\sqrt{11930}}^{1})}{13689} + 424$

Step 4.2.1.5.1.4.8

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

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 i^{2} {\sqrt{11930}}^{1 + 1}}{13689} + 424$

Step 4.2.1.5.1.4.9

Add $1$ and $1$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 i^{2} {\sqrt{11930}}^{2}}{13689} + 424$

Step 4.2.1.5.1.5

Rewrite $i^{2}$ as $- 1$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} + 4 \cdot - 1 {\sqrt{11930}}^{2}}{13689} + 424$

Step 4.2.1.5.1.6

Multiply $4$ by $- 1$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 4 {\sqrt{11930}}^{2}}{13689} + 424$

Step 4.2.1.5.1.7

Rewrite ${\sqrt{11930}}^{2}$ as $11930$ .

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Step 4.2.1.5.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11930}$ as $11930^{\frac{1}{2}}$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 4 {(11930^{\frac{1}{2}})}^{2}}{13689} + 424$

Step 4.2.1.5.1.7.2

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

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 4 \cdot 11930^{\frac{1}{2} \cdot 2}}{13689} + 424$

Step 4.2.1.5.1.7.3

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

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 4 \cdot 11930^{\frac{2}{2}}}{13689} + 424$

Step 4.2.1.5.1.7.4

Cancel the common factor of $2$ .

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Step 4.2.1.5.1.7.4.1

Cancel the common factor.

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 4 \cdot 11930^{\frac{2}{2}}}{13689} + 424$

Step 4.2.1.5.1.7.4.2

Rewrite the expression.

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 4 \cdot 11930^{1}}{13689} + 424$

Step 4.2.1.5.1.7.5

Evaluate the exponent.

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 4 \cdot 11930}{13689} + 424$

Step 4.2.1.5.1.8

Multiply $- 4$ by $11930$ .

$p = 118 \frac{16 - 8 i \sqrt{11930} - 8 i \sqrt{11930} - 47720}{13689} + 424$

Step 4.2.1.5.2

Subtract $47720$ from $16$ .

$p = 118 \frac{- 8 i \sqrt{11930} - 8 i \sqrt{11930} - 47704}{13689} + 424$

Step 4.2.1.5.3

Subtract $8 i \sqrt{11930}$ from $- 8 i \sqrt{11930}$ .

$p = 118 \frac{- 16 i \sqrt{11930} - 47704}{13689} + 424$

Step 4.2.1.6

Reorder $- 16 i \sqrt{11930}$ and $- 47704$ .

$p = 118 \frac{- 47704 - 16 i \sqrt{11930}}{13689} + 424$

Step 4.2.1.7

Combine $118$ and $\frac{- 47704 - 16 i \sqrt{11930}}{13689}$ .

$p = \frac{118 (- 47704 - 16 i \sqrt{11930})}{13689} + 424$

Step 4.2.2

To write $424$ as a fraction with a common denominator, multiply by $\frac{13689}{13689}$ .

$p = \frac{118 (- 47704 - 16 i \sqrt{11930})}{13689} + 424 \cdot \frac{13689}{13689}$

Step 4.2.3

Combine $424$ and $\frac{13689}{13689}$ .

$p = \frac{118 (- 47704 - 16 i \sqrt{11930})}{13689} + \frac{424 \cdot 13689}{13689}$

Step 4.2.4

Combine the numerators over the common denominator.

$p = \frac{175064 - 1888 i \sqrt{11930}}{13689}$

Step 5

The solution of the system of equations is all the values that make the system true.

$p = \frac{175064 + 1888 i \sqrt{11930}}{13689}, q = \frac{4 + 2 i \sqrt{11930}}{117} p = \frac{175064 - 1888 i \sqrt{11930}}{13689}, q = \frac{4 - 2 i \sqrt{11930}}{117}$

Step 6

List all of the solutions.

$p = \frac{175064 + 1888 i \sqrt{11930}}{13689}, q = \frac{4 + 2 i \sqrt{11930}}{117}$

$p = \frac{175064 - 1888 i \sqrt{11930}}{13689}, q = \frac{4 - 2 i \sqrt{11930}}{117}$