Finite Math Examples

$y = x^{2} + 2 x - 3$ , $\frac{1}{3} x - y = 5$

Step 1

Replace all occurrences of $y$ with $x^{2} + 2 x - 3$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $\frac{1}{3} x - y = 5$ with $x^{2} + 2 x - 3$ .

$\frac{1}{3} x - (x^{2} + 2 x - 3) = 5$

$y = x^{2} + 2 x - 3$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $\frac{1}{3} x - (x^{2} + 2 x - 3)$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

$\frac{x}{3} - (x^{2} + 2 x - 3) = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.1.2

Apply the distributive property.

$\frac{x}{3} - x^{2} - (2 x) + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.1.3

Simplify.

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Step 1.2.1.1.3.1

Multiply $2$ by $- 1$ .

$\frac{x}{3} - x^{2} - 2 x + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.1.3.2

Multiply $- 1$ by $- 3$ .

$\frac{x}{3} - x^{2} - 2 x + 3 = 5$

$y = x^{2} + 2 x - 3$

$\frac{x}{3} - x^{2} - 2 x + 3 = 5$

$y = x^{2} + 2 x - 3$

$\frac{x}{3} - x^{2} - 2 x + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.2

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- x^{2} + \frac{x}{3} - 2 x \cdot \frac{3}{3} + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.3

Combine $- 2 x$ and $\frac{3}{3}$ .

$- x^{2} + \frac{x}{3} + \frac{- 2 x \cdot 3}{3} + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.4

Combine the numerators over the common denominator.

$- x^{2} + \frac{x - 2 x \cdot 3}{3} + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.5

Find the common denominator.

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Step 1.2.1.5.1

Write $- x^{2}$ as a fraction with denominator $1$ .

$\frac{- x^{2}}{1} + \frac{x - 2 x \cdot 3}{3} + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.5.2

Multiply $\frac{- x^{2}}{1}$ by $\frac{3}{3}$ .

$\frac{- x^{2}}{1} \cdot \frac{3}{3} + \frac{x - 2 x \cdot 3}{3} + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.5.3

Multiply $\frac{- x^{2}}{1}$ by $\frac{3}{3}$ .

$\frac{- x^{2} \cdot 3}{3} + \frac{x - 2 x \cdot 3}{3} + 3 = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.5.4

Write $3$ as a fraction with denominator $1$ .

$\frac{- x^{2} \cdot 3}{3} + \frac{x - 2 x \cdot 3}{3} + \frac{3}{1} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.5.5

Multiply $\frac{3}{1}$ by $\frac{3}{3}$ .

$\frac{- x^{2} \cdot 3}{3} + \frac{x - 2 x \cdot 3}{3} + \frac{3}{1} \cdot \frac{3}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.5.6

Multiply $\frac{3}{1}$ by $\frac{3}{3}$ .

$\frac{- x^{2} \cdot 3}{3} + \frac{x - 2 x \cdot 3}{3} + \frac{3 \cdot 3}{3} = 5$

$y = x^{2} + 2 x - 3$

$\frac{- x^{2} \cdot 3}{3} + \frac{x - 2 x \cdot 3}{3} + \frac{3 \cdot 3}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.6

Combine the numerators over the common denominator.

$\frac{- x^{2} \cdot 3 + x - 2 x \cdot 3 + 3 \cdot 3}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.7

Simplify each term.

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Step 1.2.1.7.1

Multiply $3$ by $- 1$ .

$\frac{- 3 x^{2} + x - 2 x \cdot 3 + 3 \cdot 3}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.7.2

Multiply $3$ by $- 2$ .

$\frac{- 3 x^{2} + x - 6 x + 3 \cdot 3}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.7.3

Multiply $3$ by $3$ .

$\frac{- 3 x^{2} + x - 6 x + 9}{3} = 5$

$y = x^{2} + 2 x - 3$

$\frac{- 3 x^{2} + x - 6 x + 9}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.8

Simplify terms.

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Step 1.2.1.8.1

Subtract $6 x$ from $x$ .

$\frac{- 3 x^{2} - 5 x + 9}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.8.2

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{- (3 x^{2}) - 5 x + 9}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.8.3

Factor $- 1$ out of $- 5 x$ .

$\frac{- (3 x^{2}) - (5 x) + 9}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.8.4

Factor $- 1$ out of $- (3 x^{2}) - (5 x)$ .

$\frac{- (3 x^{2} + 5 x) + 9}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.8.5

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (3 x^{2} + 5 x) - 1 \cdot - 9}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.8.6

Factor $- 1$ out of $- (3 x^{2} + 5 x) - 1 (- 9)$ .

$\frac{- (3 x^{2} + 5 x - 9)}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.8.7

Simplify the expression.

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Step 1.2.1.8.7.1

Rewrite $- (3 x^{2} + 5 x - 9)$ as $- 1 (3 x^{2} + 5 x - 9)$ .

$\frac{- 1 (3 x^{2} + 5 x - 9)}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 1.2.1.8.7.2

Move the negative in front of the fraction.

$- \frac{3 x^{2} + 5 x - 9}{3} = 5$

$y = x^{2} + 2 x - 3$

$- \frac{3 x^{2} + 5 x - 9}{3} = 5$

$y = x^{2} + 2 x - 3$

$- \frac{3 x^{2} + 5 x - 9}{3} = 5$

$y = x^{2} + 2 x - 3$

$- \frac{3 x^{2} + 5 x - 9}{3} = 5$

$y = x^{2} + 2 x - 3$

$- \frac{3 x^{2} + 5 x - 9}{3} = 5$

$y = x^{2} + 2 x - 3$

$- \frac{3 x^{2} + 5 x - 9}{3} = 5$

$y = x^{2} + 2 x - 3$

Step 2

Solve for $x$ in $- \frac{3 x^{2} + 5 x - 9}{3} = 5$ .

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Step 2.1

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{3 x^{2} + 5 x - 9}{3}) = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $- 3 (- \frac{3 x^{2} + 5 x - 9}{3})$ .

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Step 2.2.1.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1.1

Move the leading negative in $- \frac{3 x^{2} + 5 x - 9}{3}$ into the numerator.

$- 3 \frac{- (3 x^{2} + 5 x - 9)}{3} = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

Step 2.2.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (3 x^{2} + 5 x - 9)}{3}) = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

Step 2.2.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- (3 x^{2} + 5 x - 9)}{3}) = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

Step 2.2.1.1.1.4

Rewrite the expression.

$3 x^{2} + 5 x - 9 = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

$3 x^{2} + 5 x - 9 = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

Step 2.2.1.1.2

Multiply.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (3 x^{2} + 5 x - 9) = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

Step 2.2.1.1.2.2

Multiply $3 x^{2} + 5 x - 9$ by $1$ .

$3 x^{2} + 5 x - 9 = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

$3 x^{2} + 5 x - 9 = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

$3 x^{2} + 5 x - 9 = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

$3 x^{2} + 5 x - 9 = - 3 \cdot 5$

$y = x^{2} + 2 x - 3$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 3$ by $5$ .

$3 x^{2} + 5 x - 9 = - 15$

$y = x^{2} + 2 x - 3$

$3 x^{2} + 5 x - 9 = - 15$

$y = x^{2} + 2 x - 3$

$3 x^{2} + 5 x - 9 = - 15$

$y = x^{2} + 2 x - 3$

Step 2.3

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

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Step 2.3.1

Add $15$ to both sides of the equation.

$3 x^{2} + 5 x - 9 + 15 = 0$

$y = x^{2} + 2 x - 3$

Step 2.3.2

Add $- 9$ and $15$ .

$3 x^{2} + 5 x + 6 = 0$

$y = x^{2} + 2 x - 3$

$3 x^{2} + 5 x + 6 = 0$

$y = x^{2} + 2 x - 3$

Step 2.4

Use the quadratic formula to find the solutions.

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

$y = x^{2} + 2 x - 3$

Step 2.5

Substitute the values $a = 3$ , $b = 5$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (3 \cdot 6)}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 6}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot 6$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 6}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.6.1.2.2

Multiply $- 12$ by $6$ .

$x = \frac{- 5 \pm \sqrt{25 - 72}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

$x = \frac{- 5 \pm \sqrt{25 - 72}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.6.1.3

Subtract $72$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.6.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.6.1.6

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

$x = \frac{- 5 \pm i \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

$x = \frac{- 5 \pm i \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

$x = \frac{- 5 \pm i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

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 $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 6}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.7.1.2

Multiply $- 4 \cdot 3 \cdot 6$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 6}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.7.1.2.2

Multiply $- 12$ by $6$ .

$x = \frac{- 5 \pm \sqrt{25 - 72}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

$x = \frac{- 5 \pm \sqrt{25 - 72}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.7.1.3

Subtract $72$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.7.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.7.1.6

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

$x = \frac{- 5 \pm i \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

$x = \frac{- 5 \pm i \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.7.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 2.7.3

Change the $\pm$ to $+$ .

$x = \frac{- 5 + i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 2.7.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 2.7.5

Factor $- 1$ out of $i \sqrt{47}$ .

$x = \frac{- 1 \cdot 5 - (- i \sqrt{47})}{6}$

$y = x^{2} + 2 x - 3$

Step 2.7.6

Factor $- 1$ out of $- 1 (5) - (- i \sqrt{47})$ .

$x = \frac{- 1 (5 - i \sqrt{47})}{6}$

$y = x^{2} + 2 x - 3$

Step 2.7.7

Move the negative in front of the fraction.

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 3 \cdot 6}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.8.1.2

Multiply $- 4 \cdot 3 \cdot 6$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 12 \cdot 6}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.8.1.2.2

Multiply $- 12$ by $6$ .

$x = \frac{- 5 \pm \sqrt{25 - 72}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

$x = \frac{- 5 \pm \sqrt{25 - 72}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.8.1.3

Subtract $72$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.8.1.4

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.8.1.5

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.8.1.6

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

$x = \frac{- 5 \pm i \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

$x = \frac{- 5 \pm i \sqrt{47}}{2 \cdot 3}$

$y = x^{2} + 2 x - 3$

Step 2.8.2

Multiply $2$ by $3$ .

$x = \frac{- 5 \pm i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 2.8.3

Change the $\pm$ to $-$ .

$x = \frac{- 5 - i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 2.8.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 2.8.5

Factor $- 1$ out of $- i \sqrt{47}$ .

$x = \frac{- 1 \cdot 5 - (i \sqrt{47})}{6}$

$y = x^{2} + 2 x - 3$

Step 2.8.6

Factor $- 1$ out of $- 1 (5) - (i \sqrt{47})$ .

$x = \frac{- 1 (5 + i \sqrt{47})}{6}$

$y = x^{2} + 2 x - 3$

Step 2.8.7

Move the negative in front of the fraction.

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 2.9

The final answer is the combination of both solutions.

$x = - \frac{5 - i \sqrt{47}}{6}, - \frac{5 + i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

$x = - \frac{5 - i \sqrt{47}}{6}, - \frac{5 + i \sqrt{47}}{6}$

$y = x^{2} + 2 x - 3$

Step 3

Replace all occurrences of $x$ with $- \frac{5 - i \sqrt{47}}{6}$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = x^{2} + 2 x - 3$ with $- \frac{5 - i \sqrt{47}}{6}$ .

$y = {(- \frac{5 - i \sqrt{47}}{6})}^{2} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(- \frac{5 - i \sqrt{47}}{6})}^{2} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.1.1.1

Apply the product rule to $- \frac{5 - i \sqrt{47}}{6}$ .

$y = {(- 1)}^{2} {(\frac{5 - i \sqrt{47}}{6})}^{2} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.1.2

Apply the product rule to $\frac{5 - i \sqrt{47}}{6}$ .

$y = {(- 1)}^{2} (\frac{{(5 - i \sqrt{47})}^{2}}{6^{2}}) + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = {(- 1)}^{2} (\frac{{(5 - i \sqrt{47})}^{2}}{6^{2}}) + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.2

Raise $- 1$ to the power of $2$ .

$y = 1 (\frac{{(5 - i \sqrt{47})}^{2}}{6^{2}}) + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.3

Multiply $\frac{{(5 - i \sqrt{47})}^{2}}{6^{2}}$ by $1$ .

$y = \frac{{(5 - i \sqrt{47})}^{2}}{6^{2}} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.4

Raise $6$ to the power of $2$ .

$y = \frac{{(5 - i \sqrt{47})}^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.5

Rewrite ${(5 - i \sqrt{47})}^{2}$ as $(5 - i \sqrt{47}) (5 - i \sqrt{47})$ .

$y = \frac{(5 - i \sqrt{47}) (5 - i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.6

Expand $(5 - i \sqrt{47}) (5 - i \sqrt{47})$ using the FOIL Method.

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Step 3.2.1.1.6.1

Apply the distributive property.

$y = \frac{5 (5 - i \sqrt{47}) - i \sqrt{47} (5 - i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.6.2

Apply the distributive property.

$y = \frac{5 \cdot 5 + 5 (- i \sqrt{47}) - i \sqrt{47} (5 - i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.6.3

Apply the distributive property.

$y = \frac{5 \cdot 5 + 5 (- i \sqrt{47}) - i \sqrt{47} \cdot 5 - i \sqrt{47} (- i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{5 \cdot 5 + 5 (- i \sqrt{47}) - i \sqrt{47} \cdot 5 - i \sqrt{47} (- i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7

Simplify and combine like terms.

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Step 3.2.1.1.7.1

Simplify each term.

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Step 3.2.1.1.7.1.1

Multiply $5$ by $5$ .

$y = \frac{25 + 5 (- i \sqrt{47}) - i \sqrt{47} \cdot 5 - i \sqrt{47} (- i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.2

Multiply $- 1$ by $5$ .

$y = \frac{25 - 5 (i \sqrt{47}) - i \sqrt{47} \cdot 5 - i \sqrt{47} (- i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.3

Multiply $5$ by $- 1$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} - i \sqrt{47} (- i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4

Multiply $- i \sqrt{47} (- i \sqrt{47})$ .

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Step 3.2.1.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 1 i \sqrt{47} (i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.2

Multiply $\sqrt{47}$ by $1$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + \sqrt{47} i (i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.3

Raise $\sqrt{47}$ to the power of $1$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + \sqrt{47} \sqrt{47} i i}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.4

Raise $\sqrt{47}$ to the power of $1$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + \sqrt{47} \sqrt{47} i i}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.5

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

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + {\sqrt{47}}^{1 + 1} i i}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.6

Add $1$ and $1$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + {\sqrt{47}}^{2} i i}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.7

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

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + {\sqrt{47}}^{2} (i i)}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.8

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

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + {\sqrt{47}}^{2} (i i)}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.9

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

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + {\sqrt{47}}^{2} i^{1 + 1}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.4.10

Add $1$ and $1$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + {\sqrt{47}}^{2} i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + {\sqrt{47}}^{2} i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.5

Rewrite ${\sqrt{47}}^{2}$ as $47$ .

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Step 3.2.1.1.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{47}$ as $47^{\frac{1}{2}}$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + {(47^{\frac{1}{2}})}^{2} i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.5.2

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

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 47^{\frac{1}{2} \cdot 2} i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.5.3

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

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 47^{\frac{2}{2}} i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 3.2.1.1.7.1.5.4.1

Cancel the common factor.

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 47^{\frac{2}{2}} i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.5.4.2

Rewrite the expression.

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 47 i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 47 i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.5.5

Evaluate the exponent.

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 47 i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 47 i^{2}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.6

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

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} + 47 \cdot - 1}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.1.7

Multiply $47$ by $- 1$ .

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} - 47}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{25 - 5 i \sqrt{47} - 5 i \sqrt{47} - 47}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.2

Subtract $47$ from $25$ .

$y = \frac{- 5 i \sqrt{47} - 5 i \sqrt{47} - 22}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.7.3

Subtract $5 i \sqrt{47}$ from $- 5 i \sqrt{47}$ .

$y = \frac{- 10 i \sqrt{47} - 22}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{- 10 i \sqrt{47} - 22}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.8

Reorder $- 10 i \sqrt{47}$ and $- 22$ .

$y = \frac{- 22 - 10 i \sqrt{47}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.9

Cancel the common factor of $- 22 - 10 i \sqrt{47}$ and $36$ .

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Step 3.2.1.1.9.1

Factor $2$ out of $- 22$ .

$y = \frac{2 (- 11) - 10 i \sqrt{47}}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.9.2

Factor $2$ out of $- 10 i \sqrt{47}$ .

$y = \frac{2 (- 11) + 2 (- 5 i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.9.3

Factor $2$ out of $2 (- 11) + 2 (- 5 i \sqrt{47})$ .

$y = \frac{2 (- 11 - 5 i \sqrt{47})}{36} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.9.4

Cancel the common factors.

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Step 3.2.1.1.9.4.1

Factor $2$ out of $36$ .

$y = \frac{2 (- 11 - 5 i \sqrt{47})}{2 \cdot 18} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.9.4.2

Cancel the common factor.

$y = \frac{2 (- 11 - 5 i \sqrt{47})}{2 \cdot 18} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.9.4.3

Rewrite the expression.

$y = \frac{- 11 - 5 i \sqrt{47}}{18} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{- 11 - 5 i \sqrt{47}}{18} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{- 11 - 5 i \sqrt{47}}{18} + 2 (- \frac{5 - i \sqrt{47}}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.10

Cancel the common factor of $2$ .

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Step 3.2.1.1.10.1

Move the leading negative in $- \frac{5 - i \sqrt{47}}{6}$ into the numerator.

$y = \frac{- 11 - 5 i \sqrt{47}}{18} + 2 (\frac{- (5 - i \sqrt{47})}{6}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.10.2

Factor $2$ out of $6$ .

$y = \frac{- 11 - 5 i \sqrt{47}}{18} + 2 (\frac{- (5 - i \sqrt{47})}{2 (3)}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.10.3

Cancel the common factor.

$y = \frac{- 11 - 5 i \sqrt{47}}{18} + 2 (\frac{- (5 - i \sqrt{47})}{2 \cdot 3}) - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.10.4

Rewrite the expression.

$y = \frac{- 11 - 5 i \sqrt{47}}{18} + \frac{- (5 - i \sqrt{47})}{3} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{- 11 - 5 i \sqrt{47}}{18} + \frac{- (5 - i \sqrt{47})}{3} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.1.11

Move the negative in front of the fraction.

$y = \frac{- 11 - 5 i \sqrt{47}}{18} - \frac{5 - i \sqrt{47}}{3} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{- 11 - 5 i \sqrt{47}}{18} - \frac{5 - i \sqrt{47}}{3} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.2

To write $- \frac{5 - i \sqrt{47}}{3}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = \frac{- 11 - 5 i \sqrt{47}}{18} - \frac{5 - i \sqrt{47}}{3} \cdot \frac{6}{6} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.3

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.1.3.1

Multiply $\frac{5 - i \sqrt{47}}{3}$ by $\frac{6}{6}$ .

$y = \frac{- 11 - 5 i \sqrt{47}}{18} - \frac{(5 - i \sqrt{47}) \cdot 6}{3 \cdot 6} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.3.2

Multiply $3$ by $6$ .

$y = \frac{- 11 - 5 i \sqrt{47}}{18} - \frac{(5 - i \sqrt{47}) \cdot 6}{18} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = \frac{- 11 - 5 i \sqrt{47}}{18} - \frac{(5 - i \sqrt{47}) \cdot 6}{18} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.4

Combine the numerators over the common denominator.

$y = \frac{- 41 + i \sqrt{47}}{18} - 3$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.5

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$y = \frac{- 41 + i \sqrt{47}}{18} - 3 \cdot \frac{18}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.6

Combine $- 3$ and $\frac{18}{18}$ .

$y = \frac{- 41 + i \sqrt{47}}{18} + \frac{- 3 \cdot 18}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.7

Combine the numerators over the common denominator.

$y = \frac{- 95 + i \sqrt{47}}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.8

Rewrite $- 95$ as $- 1 (95)$ .

$y = \frac{- 1 \cdot 95 + i \sqrt{47}}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.9

Factor $- 1$ out of $i \sqrt{47}$ .

$y = \frac{- 1 \cdot 95 - (- i \sqrt{47})}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.10

Factor $- 1$ out of $- 1 (95) - (- i \sqrt{47})$ .

$y = \frac{- 1 (95 - i \sqrt{47})}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 3.2.1.11

Move the negative in front of the fraction.

$y = - \frac{95 - i \sqrt{47}}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = - \frac{95 - i \sqrt{47}}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = - \frac{95 - i \sqrt{47}}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

$y = - \frac{95 - i \sqrt{47}}{18}$

$x = - \frac{5 - i \sqrt{47}}{6}$

Step 4

Replace all occurrences of $x$ with $- \frac{5 + i \sqrt{47}}{6}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = x^{2} + 2 x - 3$ with $- \frac{5 + i \sqrt{47}}{6}$ .

$y = {(- \frac{5 + i \sqrt{47}}{6})}^{2} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(- \frac{5 + i \sqrt{47}}{6})}^{2} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.1.1.1.1

Apply the product rule to $- \frac{5 + i \sqrt{47}}{6}$ .

$y = {(- 1)}^{2} {(\frac{5 + i \sqrt{47}}{6})}^{2} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.1.2

Apply the product rule to $\frac{5 + i \sqrt{47}}{6}$ .

$y = {(- 1)}^{2} (\frac{{(5 + i \sqrt{47})}^{2}}{6^{2}}) + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = {(- 1)}^{2} (\frac{{(5 + i \sqrt{47})}^{2}}{6^{2}}) + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$y = 1 (\frac{{(5 + i \sqrt{47})}^{2}}{6^{2}}) + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.3

Multiply $\frac{{(5 + i \sqrt{47})}^{2}}{6^{2}}$ by $1$ .

$y = \frac{{(5 + i \sqrt{47})}^{2}}{6^{2}} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.4

Raise $6$ to the power of $2$ .

$y = \frac{{(5 + i \sqrt{47})}^{2}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.5

Rewrite ${(5 + i \sqrt{47})}^{2}$ as $(5 + i \sqrt{47}) (5 + i \sqrt{47})$ .

$y = \frac{(5 + i \sqrt{47}) (5 + i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.6

Expand $(5 + i \sqrt{47}) (5 + i \sqrt{47})$ using the FOIL Method.

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Step 4.2.1.1.6.1

Apply the distributive property.

$y = \frac{5 (5 + i \sqrt{47}) + i \sqrt{47} (5 + i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.6.2

Apply the distributive property.

$y = \frac{5 \cdot 5 + 5 (i \sqrt{47}) + i \sqrt{47} (5 + i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.6.3

Apply the distributive property.

$y = \frac{5 \cdot 5 + 5 (i \sqrt{47}) + i \sqrt{47} \cdot 5 + i \sqrt{47} (i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{5 \cdot 5 + 5 (i \sqrt{47}) + i \sqrt{47} \cdot 5 + i \sqrt{47} (i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7

Simplify and combine like terms.

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Step 4.2.1.1.7.1

Simplify each term.

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Step 4.2.1.1.7.1.1

Multiply $5$ by $5$ .

$y = \frac{25 + 5 (i \sqrt{47}) + i \sqrt{47} \cdot 5 + i \sqrt{47} (i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.2

Move $5$ to the left of $i \sqrt{47}$ .

$y = \frac{25 + 5 i \sqrt{47} + 5 \cdot (i \sqrt{47}) + i \sqrt{47} (i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.3

Multiply $i \sqrt{47} (i \sqrt{47})$ .

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Step 4.2.1.1.7.1.3.1

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

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i i \sqrt{47} \sqrt{47}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.3.2

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

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i i \sqrt{47} \sqrt{47}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.3.3

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

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i^{1 + 1} \sqrt{47} \sqrt{47}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.3.4

Add $1$ and $1$ .

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i^{2} \sqrt{47} \sqrt{47}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.3.5

Raise $\sqrt{47}$ to the power of $1$ .

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i^{2} (\sqrt{47} \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.3.6

Raise $\sqrt{47}$ to the power of $1$ .

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i^{2} (\sqrt{47} \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.3.7

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

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i^{2} {\sqrt{47}}^{1 + 1}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.3.8

Add $1$ and $1$ .

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i^{2} {\sqrt{47}}^{2}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} + i^{2} {\sqrt{47}}^{2}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.4

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

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 {\sqrt{47}}^{2}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.5

Rewrite ${\sqrt{47}}^{2}$ as $47$ .

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Step 4.2.1.1.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{47}$ as $47^{\frac{1}{2}}$ .

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 {(47^{\frac{1}{2}})}^{2}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.5.2

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

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 \cdot 47^{\frac{1}{2} \cdot 2}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.5.3

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

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 \cdot 47^{\frac{2}{2}}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.1.1.7.1.5.4.1

Cancel the common factor.

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 \cdot 47^{\frac{2}{2}}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.5.4.2

Rewrite the expression.

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 \cdot 47}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 \cdot 47}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.5.5

Evaluate the exponent.

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 \cdot 47}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 1 \cdot 47}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.1.6

Multiply $- 1$ by $47$ .

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 47}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{25 + 5 i \sqrt{47} + 5 i \sqrt{47} - 47}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.2

Subtract $47$ from $25$ .

$y = \frac{5 i \sqrt{47} + 5 i \sqrt{47} - 22}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.7.3

Add $5 i \sqrt{47}$ and $5 i \sqrt{47}$ .

$y = \frac{10 i \sqrt{47} - 22}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{10 i \sqrt{47} - 22}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.8

Reorder $10 i \sqrt{47}$ and $- 22$ .

$y = \frac{- 22 + 10 i \sqrt{47}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.9

Cancel the common factor of $- 22 + 10 i \sqrt{47}$ and $36$ .

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Step 4.2.1.1.9.1

Factor $2$ out of $- 22$ .

$y = \frac{2 (- 11) + 10 i \sqrt{47}}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.9.2

Factor $2$ out of $10 i \sqrt{47}$ .

$y = \frac{2 (- 11) + 2 (5 i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.9.3

Factor $2$ out of $2 (- 11) + 2 (5 i \sqrt{47})$ .

$y = \frac{2 (- 11 + 5 i \sqrt{47})}{36} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.9.4

Cancel the common factors.

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Step 4.2.1.1.9.4.1

Factor $2$ out of $36$ .

$y = \frac{2 (- 11 + 5 i \sqrt{47})}{2 \cdot 18} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.9.4.2

Cancel the common factor.

$y = \frac{2 (- 11 + 5 i \sqrt{47})}{2 \cdot 18} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.9.4.3

Rewrite the expression.

$y = \frac{- 11 + 5 i \sqrt{47}}{18} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{- 11 + 5 i \sqrt{47}}{18} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{- 11 + 5 i \sqrt{47}}{18} + 2 (- \frac{5 + i \sqrt{47}}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.10

Cancel the common factor of $2$ .

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Step 4.2.1.1.10.1

Move the leading negative in $- \frac{5 + i \sqrt{47}}{6}$ into the numerator.

$y = \frac{- 11 + 5 i \sqrt{47}}{18} + 2 (\frac{- (5 + i \sqrt{47})}{6}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.10.2

Factor $2$ out of $6$ .

$y = \frac{- 11 + 5 i \sqrt{47}}{18} + 2 (\frac{- (5 + i \sqrt{47})}{2 (3)}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.10.3

Cancel the common factor.

$y = \frac{- 11 + 5 i \sqrt{47}}{18} + 2 (\frac{- (5 + i \sqrt{47})}{2 \cdot 3}) - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.10.4

Rewrite the expression.

$y = \frac{- 11 + 5 i \sqrt{47}}{18} + \frac{- (5 + i \sqrt{47})}{3} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{- 11 + 5 i \sqrt{47}}{18} + \frac{- (5 + i \sqrt{47})}{3} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.1.11

Move the negative in front of the fraction.

$y = \frac{- 11 + 5 i \sqrt{47}}{18} - \frac{5 + i \sqrt{47}}{3} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{- 11 + 5 i \sqrt{47}}{18} - \frac{5 + i \sqrt{47}}{3} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.2

To write $- \frac{5 + i \sqrt{47}}{3}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = \frac{- 11 + 5 i \sqrt{47}}{18} - \frac{5 + i \sqrt{47}}{3} \cdot \frac{6}{6} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.3

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{5 + i \sqrt{47}}{3}$ by $\frac{6}{6}$ .

$y = \frac{- 11 + 5 i \sqrt{47}}{18} - \frac{(5 + i \sqrt{47}) \cdot 6}{3 \cdot 6} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.3.2

Multiply $3$ by $6$ .

$y = \frac{- 11 + 5 i \sqrt{47}}{18} - \frac{(5 + i \sqrt{47}) \cdot 6}{18} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = \frac{- 11 + 5 i \sqrt{47}}{18} - \frac{(5 + i \sqrt{47}) \cdot 6}{18} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$y = \frac{- 41 - i \sqrt{47}}{18} - 3$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.5

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{18}{18}$ .

$y = \frac{- 41 - i \sqrt{47}}{18} - 3 \cdot \frac{18}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.6

Combine $- 3$ and $\frac{18}{18}$ .

$y = \frac{- 41 - i \sqrt{47}}{18} + \frac{- 3 \cdot 18}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.7

Combine the numerators over the common denominator.

$y = \frac{- 95 - i \sqrt{47}}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.8

Rewrite $- 95$ as $- 1 (95)$ .

$y = \frac{- 1 \cdot 95 - i \sqrt{47}}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.9

Factor $- 1$ out of $- i \sqrt{47}$ .

$y = \frac{- 1 \cdot 95 - (i \sqrt{47})}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.10

Factor $- 1$ out of $- 1 (95) - (i \sqrt{47})$ .

$y = \frac{- 1 (95 + i \sqrt{47})}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 4.2.1.11

Move the negative in front of the fraction.

$y = - \frac{95 + i \sqrt{47}}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = - \frac{95 + i \sqrt{47}}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = - \frac{95 + i \sqrt{47}}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

$y = - \frac{95 + i \sqrt{47}}{18}$

$x = - \frac{5 + i \sqrt{47}}{6}$

Step 5

List all of the solutions.

$y = - \frac{95 - i \sqrt{47}}{18}, x = - \frac{5 - i \sqrt{47}}{6}$

$y = - \frac{95 + i \sqrt{47}}{18}, x = - \frac{5 + i \sqrt{47}}{6}$

Step 6