Algebra Examples

$2 x - y = 4$ ${(x + 3)}^{2} + y^{2} = 8$

Step 1

Solve for $x$ in $2 x - y = 4$ .

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Step 1.1

Add $y$ to both sides of the equation.

$2 x = 4 + y$

${(x + 3)}^{2} + y^{2} = 8$

Step 1.2

Divide each term in $2 x = 4 + y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 4 + y$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2} + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2} + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{2} + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

$x = \frac{4}{2} + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

$x = \frac{4}{2} + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $4$ by $2$ .

$x = 2 + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

$x = 2 + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

$x = 2 + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

$x = 2 + \frac{y}{2}$

${(x + 3)}^{2} + y^{2} = 8$

Step 2

Replace all occurrences of $x$ with $2 + \frac{y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in ${(x + 3)}^{2} + y^{2} = 8$ with $2 + \frac{y}{2}$ .

${((2 + \frac{y}{2}) + 3)}^{2} + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${((2 + \frac{y}{2}) + 3)}^{2} + y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Add $2$ and $3$ .

${(\frac{y}{2} + 5)}^{2} + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.2

Rewrite ${(\frac{y}{2} + 5)}^{2}$ as $(\frac{y}{2} + 5) (\frac{y}{2} + 5)$ .

$(\frac{y}{2} + 5) (\frac{y}{2} + 5) + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.3

Expand $(\frac{y}{2} + 5) (\frac{y}{2} + 5)$ using the FOIL Method.

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Step 2.2.1.1.3.1

Apply the distributive property.

$\frac{y}{2} \cdot (\frac{y}{2} + 5) + 5 (\frac{y}{2} + 5) + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.3.2

Apply the distributive property.

$\frac{y}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2} + 5) + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.3.3

Apply the distributive property.

$\frac{y}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

$\frac{y}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4

Simplify and combine like terms.

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Step 2.2.1.1.4.1

Simplify each term.

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Step 2.2.1.1.4.1.1

Multiply $\frac{y}{2} \cdot \frac{y}{2}$ .

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Step 2.2.1.1.4.1.1.1

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

$\frac{y \cdot y}{2 \cdot 2} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.1.2

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

$\frac{y \cdot y}{2 \cdot 2} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.1.3

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

$\frac{y \cdot y}{2 \cdot 2} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.1.4

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

$\frac{y^{1 + 1}}{2 \cdot 2} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.1.5

Add $1$ and $1$ .

$\frac{y^{2}}{2 \cdot 2} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.1.6

Multiply $2$ by $2$ .

$\frac{y^{2}}{4} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

$\frac{y^{2}}{4} + \frac{y}{2} \cdot 5 + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.2

Combine $\frac{y}{2}$ and $5$ .

$\frac{y^{2}}{4} + \frac{y \cdot 5}{2} + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.3

Move $5$ to the left of $y$ .

$\frac{y^{2}}{4} + \frac{5 \cdot y}{2} + 5 (\frac{y}{2}) + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.4

Combine $5$ and $\frac{y}{2}$ .

$\frac{y^{2}}{4} + \frac{5 y}{2} + \frac{5 y}{2} + 5 \cdot 5 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.1.5

Multiply $5$ by $5$ .

$\frac{y^{2}}{4} + \frac{5 y}{2} + \frac{5 y}{2} + 25 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

$\frac{y^{2}}{4} + \frac{5 y}{2} + \frac{5 y}{2} + 25 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.4.2

Add $\frac{5 y}{2}$ and $\frac{5 y}{2}$ .

$\frac{y^{2}}{4} + 2 (\frac{5 y}{2}) + 25 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

$\frac{y^{2}}{4} + 2 (\frac{5 y}{2}) + 25 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.5

Cancel the common factor of $2$ .

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Step 2.2.1.1.5.1

Cancel the common factor.

$\frac{y^{2}}{4} + 2 (\frac{5 y}{2}) + 25 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.1.5.2

Rewrite the expression.

$\frac{y^{2}}{4} + 5 y + 25 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

$\frac{y^{2}}{4} + 5 y + 25 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

$\frac{y^{2}}{4} + 5 y + 25 + y^{2} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5 y + 25 + \frac{y^{2}}{4} + y^{2} \cdot \frac{4}{4} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.3

Combine $y^{2}$ and $\frac{4}{4}$ .

$5 y + 25 + \frac{y^{2}}{4} + \frac{y^{2} \cdot 4}{4} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$5 y + 25 + \frac{y^{2} + y^{2} \cdot 4}{4} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.5

Add $y^{2}$ and $y^{2} \cdot 4$ .

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Step 2.2.1.5.1

Reorder $y^{2}$ and $4$ .

$5 y + 25 + \frac{y^{2} + 4 \cdot y^{2}}{4} = 8$

$x = 2 + \frac{y}{2}$

Step 2.2.1.5.2

Add $y^{2}$ and $4 \cdot y^{2}$ .

$5 y + 25 + \frac{5 y^{2}}{4} = 8$

$x = 2 + \frac{y}{2}$

$5 y + 25 + \frac{5 y^{2}}{4} = 8$

$x = 2 + \frac{y}{2}$

$5 y + 25 + \frac{5 y^{2}}{4} = 8$

$x = 2 + \frac{y}{2}$

$5 y + 25 + \frac{5 y^{2}}{4} = 8$

$x = 2 + \frac{y}{2}$

$5 y + 25 + \frac{5 y^{2}}{4} = 8$

$x = 2 + \frac{y}{2}$

Step 3

Solve for $y$ in $5 y + 25 + \frac{5 y^{2}}{4} = 8$ .

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Step 3.1

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

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Step 3.1.1

Subtract $8$ from both sides of the equation.

$5 y + 25 + \frac{5 y^{2}}{4} - 8 = 0$

$x = 2 + \frac{y}{2}$

Step 3.1.2

Subtract $8$ from $25$ .

$5 y + \frac{5 y^{2}}{4} + 17 = 0$

$x = 2 + \frac{y}{2}$

$5 y + \frac{5 y^{2}}{4} + 17 = 0$

$x = 2 + \frac{y}{2}$

Step 3.2

Multiply through by the least common denominator $4$ , then simplify.

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Step 3.2.1

Apply the distributive property.

$4 (5 y) + 4 (\frac{5 y^{2}}{4}) + 4 \cdot 17 = 0$

$x = 2 + \frac{y}{2}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Multiply $5$ by $4$ .

$20 y + 4 (\frac{5 y^{2}}{4}) + 4 \cdot 17 = 0$

$x = 2 + \frac{y}{2}$

Step 3.2.2.2

Cancel the common factor of $4$ .

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Step 3.2.2.2.1

Cancel the common factor.

$20 y + 4 (\frac{5 y^{2}}{4}) + 4 \cdot 17 = 0$

$x = 2 + \frac{y}{2}$

Step 3.2.2.2.2

Rewrite the expression.

$20 y + 5 y^{2} + 4 \cdot 17 = 0$

$x = 2 + \frac{y}{2}$

$20 y + 5 y^{2} + 4 \cdot 17 = 0$

$x = 2 + \frac{y}{2}$

Step 3.2.2.3

Multiply $4$ by $17$ .

$20 y + 5 y^{2} + 68 = 0$

$x = 2 + \frac{y}{2}$

$20 y + 5 y^{2} + 68 = 0$

$x = 2 + \frac{y}{2}$

Step 3.2.3

Reorder $20 y$ and $5 y^{2}$ .

$5 y^{2} + 20 y + 68 = 0$

$x = 2 + \frac{y}{2}$

$5 y^{2} + 20 y + 68 = 0$

$x = 2 + \frac{y}{2}$

Step 3.3

Use the quadratic formula to find the solutions.

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

$x = 2 + \frac{y}{2}$

Step 3.4

Substitute the values $a = 5$ , $b = 20$ , and $c = 68$ into the quadratic formula and solve for $y$ .

$\frac{- 20 \pm \sqrt{20^{2} - 4 \cdot (5 \cdot 68)}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

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 $20$ to the power of $2$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot 5 \cdot 68}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.2

Multiply $- 4 \cdot 5 \cdot 68$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{- 20 \pm \sqrt{400 - 20 \cdot 68}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.2.2

Multiply $- 20$ by $68$ .

$y = \frac{- 20 \pm \sqrt{400 - 1360}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm \sqrt{400 - 1360}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.3

Subtract $1360$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.4

Rewrite $- 960$ as $- 1 (960)$ .

$y = \frac{- 20 \pm \sqrt{- 1 \cdot 960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (960)}$ as $\sqrt{- 1} \cdot \sqrt{960}$ .

$y = \frac{- 20 \pm \sqrt{- 1} \cdot \sqrt{960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.6

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

$y = \frac{- 20 \pm i \cdot \sqrt{960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.7

Rewrite $960$ as $8^{2} \cdot 15$ .

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Step 3.5.1.7.1

Factor $64$ out of $960$ .

$y = \frac{- 20 \pm i \cdot \sqrt{64 (15)}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.7.2

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 20 \pm i \cdot \sqrt{8^{2} \cdot 15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm i \cdot \sqrt{8^{2} \cdot 15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.8

Pull terms out from under the radical.

$y = \frac{- 20 \pm i \cdot (8 \sqrt{15})}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.1.9

Move $8$ to the left of $i$ .

$y = \frac{- 20 \pm 8 i \sqrt{15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm 8 i \sqrt{15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.5.2

Multiply $2$ by $5$ .

$y = \frac{- 20 \pm 8 i \sqrt{15}}{10}$

$x = 2 + \frac{y}{2}$

Step 3.5.3

Simplify $\frac{- 20 \pm 8 i \sqrt{15}}{10}$ .

$y = \frac{- 10 \pm 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 10 \pm 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

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 $20$ to the power of $2$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot 5 \cdot 68}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.2

Multiply $- 4 \cdot 5 \cdot 68$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{- 20 \pm \sqrt{400 - 20 \cdot 68}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.2.2

Multiply $- 20$ by $68$ .

$y = \frac{- 20 \pm \sqrt{400 - 1360}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm \sqrt{400 - 1360}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.3

Subtract $1360$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.4

Rewrite $- 960$ as $- 1 (960)$ .

$y = \frac{- 20 \pm \sqrt{- 1 \cdot 960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.5

Rewrite $\sqrt{- 1 (960)}$ as $\sqrt{- 1} \cdot \sqrt{960}$ .

$y = \frac{- 20 \pm \sqrt{- 1} \cdot \sqrt{960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.6

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

$y = \frac{- 20 \pm i \cdot \sqrt{960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.7

Rewrite $960$ as $8^{2} \cdot 15$ .

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Step 3.6.1.7.1

Factor $64$ out of $960$ .

$y = \frac{- 20 \pm i \cdot \sqrt{64 (15)}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.7.2

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 20 \pm i \cdot \sqrt{8^{2} \cdot 15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm i \cdot \sqrt{8^{2} \cdot 15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.8

Pull terms out from under the radical.

$y = \frac{- 20 \pm i \cdot (8 \sqrt{15})}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.1.9

Move $8$ to the left of $i$ .

$y = \frac{- 20 \pm 8 i \sqrt{15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm 8 i \sqrt{15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.6.2

Multiply $2$ by $5$ .

$y = \frac{- 20 \pm 8 i \sqrt{15}}{10}$

$x = 2 + \frac{y}{2}$

Step 3.6.3

Simplify $\frac{- 20 \pm 8 i \sqrt{15}}{10}$ .

$y = \frac{- 10 \pm 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

Step 3.6.4

Change the $\pm$ to $+$ .

$y = \frac{- 10 + 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

Step 3.6.5

Rewrite $- 10$ as $- 1 (10)$ .

$y = \frac{- 1 \cdot 10 + 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

Step 3.6.6

Factor $- 1$ out of $4 i \sqrt{15}$ .

$y = \frac{- 1 \cdot 10 - (- 4 i \sqrt{15})}{5}$

$x = 2 + \frac{y}{2}$

Step 3.6.7

Factor $- 1$ out of $- 1 (10) - (- 4 i \sqrt{15})$ .

$y = \frac{- 1 (10 - 4 i \sqrt{15})}{5}$

$x = 2 + \frac{y}{2}$

Step 3.6.8

Move the negative in front of the fraction.

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

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 $20$ to the power of $2$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot 5 \cdot 68}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.2

Multiply $- 4 \cdot 5 \cdot 68$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{- 20 \pm \sqrt{400 - 20 \cdot 68}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.2.2

Multiply $- 20$ by $68$ .

$y = \frac{- 20 \pm \sqrt{400 - 1360}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm \sqrt{400 - 1360}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.3

Subtract $1360$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.4

Rewrite $- 960$ as $- 1 (960)$ .

$y = \frac{- 20 \pm \sqrt{- 1 \cdot 960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.5

Rewrite $\sqrt{- 1 (960)}$ as $\sqrt{- 1} \cdot \sqrt{960}$ .

$y = \frac{- 20 \pm \sqrt{- 1} \cdot \sqrt{960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.6

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

$y = \frac{- 20 \pm i \cdot \sqrt{960}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.7

Rewrite $960$ as $8^{2} \cdot 15$ .

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Step 3.7.1.7.1

Factor $64$ out of $960$ .

$y = \frac{- 20 \pm i \cdot \sqrt{64 (15)}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.7.2

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 20 \pm i \cdot \sqrt{8^{2} \cdot 15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm i \cdot \sqrt{8^{2} \cdot 15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.8

Pull terms out from under the radical.

$y = \frac{- 20 \pm i \cdot (8 \sqrt{15})}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.1.9

Move $8$ to the left of $i$ .

$y = \frac{- 20 \pm 8 i \sqrt{15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

$y = \frac{- 20 \pm 8 i \sqrt{15}}{2 \cdot 5}$

$x = 2 + \frac{y}{2}$

Step 3.7.2

Multiply $2$ by $5$ .

$y = \frac{- 20 \pm 8 i \sqrt{15}}{10}$

$x = 2 + \frac{y}{2}$

Step 3.7.3

Simplify $\frac{- 20 \pm 8 i \sqrt{15}}{10}$ .

$y = \frac{- 10 \pm 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

Step 3.7.4

Change the $\pm$ to $-$ .

$y = \frac{- 10 - 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

Step 3.7.5

Rewrite $- 10$ as $- 1 (10)$ .

$y = \frac{- 1 \cdot 10 - 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

Step 3.7.6

Factor $- 1$ out of $- 4 i \sqrt{15}$ .

$y = \frac{- 1 \cdot 10 - (4 i \sqrt{15})}{5}$

$x = 2 + \frac{y}{2}$

Step 3.7.7

Factor $- 1$ out of $- 1 (10) - (4 i \sqrt{15})$ .

$y = \frac{- 1 (10 + 4 i \sqrt{15})}{5}$

$x = 2 + \frac{y}{2}$

Step 3.7.8

Move the negative in front of the fraction.

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

Step 3.8

The final answer is the combination of both solutions.

$y = - \frac{10 - 4 i \sqrt{15}}{5}, - \frac{10 + 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}, - \frac{10 + 4 i \sqrt{15}}{5}$

$x = 2 + \frac{y}{2}$

Step 4

Replace all occurrences of $y$ with $- \frac{10 - 4 i \sqrt{15}}{5}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 2 + \frac{y}{2}$ with $- \frac{10 - 4 i \sqrt{15}}{5}$ .

$x = 2 + \frac{- \frac{10 - 4 i \sqrt{15}}{5}}{2}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 + \frac{- \frac{10 - 4 i \sqrt{15}}{5}}{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 2 - \frac{10 - 4 i \sqrt{15}}{5} \cdot \frac{1}{2}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.1.2

Multiply $- \frac{10 - 4 i \sqrt{15}}{5} \cdot \frac{1}{2}$ .

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Step 4.2.1.1.2.1

Multiply $\frac{1}{2}$ by $\frac{10 - 4 i \sqrt{15}}{5}$ .

$x = 2 - \frac{10 - 4 i \sqrt{15}}{2 \cdot 5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.1.2.2

Multiply $2$ by $5$ .

$x = 2 - \frac{10 - 4 i \sqrt{15}}{10}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = 2 - \frac{10 - 4 i \sqrt{15}}{10}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.1.3

Cancel the common factor of $10 - 4 i \sqrt{15}$ and $10$ .

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Step 4.2.1.1.3.1

Factor $2$ out of $10$ .

$x = 2 - \frac{2 (5) - 4 i \sqrt{15}}{10}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.1.3.2

Factor $2$ out of $- 4 i \sqrt{15}$ .

$x = 2 - \frac{2 (5) + 2 (- 2 i \sqrt{15})}{10}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.1.3.3

Factor $2$ out of $2 (5) + 2 (- 2 i \sqrt{15})$ .

$x = 2 - \frac{2 (5 - 2 i \sqrt{15})}{10}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.1.3.4

Cancel the common factors.

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Step 4.2.1.1.3.4.1

Factor $2$ out of $10$ .

$x = 2 - \frac{2 (5 - 2 i \sqrt{15})}{2 \cdot 5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.1.3.4.2

Cancel the common factor.

$x = 2 - \frac{2 (5 - 2 i \sqrt{15})}{2 \cdot 5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.1.3.4.3

Rewrite the expression.

$x = 2 - \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = 2 - \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = 2 - \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = 2 - \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = 2 \cdot \frac{5}{5} - \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.3

Combine $2$ and $\frac{5}{5}$ .

$x = \frac{2 \cdot 5}{5} - \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 - 4 i \sqrt{15}}{5}$

Step 5

Replace all occurrences of $y$ with $- \frac{10 + 4 i \sqrt{15}}{5}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 2 + \frac{y}{2}$ with $- \frac{10 + 4 i \sqrt{15}}{5}$ .

$x = 2 + \frac{- \frac{10 + 4 i \sqrt{15}}{5}}{2}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $2 + \frac{- \frac{10 + 4 i \sqrt{15}}{5}}{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = 2 - \frac{10 + 4 i \sqrt{15}}{5} \cdot \frac{1}{2}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.1.2

Multiply $- \frac{10 + 4 i \sqrt{15}}{5} \cdot \frac{1}{2}$ .

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Step 5.2.1.1.2.1

Multiply $\frac{1}{2}$ by $\frac{10 + 4 i \sqrt{15}}{5}$ .

$x = 2 - \frac{10 + 4 i \sqrt{15}}{2 \cdot 5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.1.2.2

Multiply $2$ by $5$ .

$x = 2 - \frac{10 + 4 i \sqrt{15}}{10}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = 2 - \frac{10 + 4 i \sqrt{15}}{10}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.1.3

Cancel the common factor of $10 + 4 i \sqrt{15}$ and $10$ .

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Step 5.2.1.1.3.1

Factor $2$ out of $10$ .

$x = 2 - \frac{2 (5) + 4 i \sqrt{15}}{10}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.1.3.2

Factor $2$ out of $4 i \sqrt{15}$ .

$x = 2 - \frac{2 (5) + 2 (2 i \sqrt{15})}{10}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.1.3.3

Factor $2$ out of $2 (5) + 2 (2 i \sqrt{15})$ .

$x = 2 - \frac{2 (5 + 2 i \sqrt{15})}{10}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.1.3.4

Cancel the common factors.

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Step 5.2.1.1.3.4.1

Factor $2$ out of $10$ .

$x = 2 - \frac{2 (5 + 2 i \sqrt{15})}{2 \cdot 5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.1.3.4.2

Cancel the common factor.

$x = 2 - \frac{2 (5 + 2 i \sqrt{15})}{2 \cdot 5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.1.3.4.3

Rewrite the expression.

$x = 2 - \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = 2 - \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = 2 - \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = 2 - \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = 2 \cdot \frac{5}{5} - \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.3

Combine $2$ and $\frac{5}{5}$ .

$x = \frac{2 \cdot 5}{5} - \frac{5 + 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x = \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

$x = \frac{5 - 2 i \sqrt{15}}{5}$

$y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 6

List all of the solutions.

$x = \frac{5 + 2 i \sqrt{15}}{5}, y = - \frac{10 - 4 i \sqrt{15}}{5}$

$x = \frac{5 - 2 i \sqrt{15}}{5}, y = - \frac{10 + 4 i \sqrt{15}}{5}$

Step 7