Algebra Examples

$4 x^{2} + 9 y^{2} = 72$ $x - y^{2} = - 1$

Step 1

Add $y^{2}$ to both sides of the equation.

$x = - 1 + y^{2}$

$4 x^{2} + 9 y^{2} = 72$

Step 2

Replace all occurrences of $x$ with $- 1 + y^{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x^{2} + 9 y^{2} = 72$ with $- 1 + y^{2}$ .

$4 {(- 1 + y^{2})}^{2} + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 {(- 1 + y^{2})}^{2} + 9 y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(- 1 + y^{2})}^{2}$ as $(- 1 + y^{2}) (- 1 + y^{2})$ .

$4 ((- 1 + y^{2}) (- 1 + y^{2})) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.2

Expand $(- 1 + y^{2}) (- 1 + y^{2})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$4 (- 1 (- 1 + y^{2}) + y^{2} (- 1 + y^{2})) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.2.2

Apply the distributive property.

$4 (- 1 \cdot - 1 - 1 y^{2} + y^{2} (- 1 + y^{2})) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.2.3

Apply the distributive property.

$4 (- 1 \cdot - 1 - 1 y^{2} + y^{2} \cdot - 1 + y^{2} y^{2}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

$4 (- 1 \cdot - 1 - 1 y^{2} + y^{2} \cdot - 1 + y^{2} y^{2}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $- 1$ by $- 1$ .

$4 (1 - 1 y^{2} + y^{2} \cdot - 1 + y^{2} y^{2}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.3.1.2

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

$4 (1 - y^{2} + y^{2} \cdot - 1 + y^{2} y^{2}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.3.1.3

Move $- 1$ to the left of $y^{2}$ .

$4 (1 - y^{2} - 1 \cdot y^{2} + y^{2} y^{2}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.3.1.4

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

$4 (1 - y^{2} - y^{2} + y^{2} y^{2}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.3.1.5

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 2.2.1.1.3.1.5.1

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

$4 (1 - y^{2} - y^{2} + y^{2 + 2}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.3.1.5.2

Add $2$ and $2$ .

$4 (1 - y^{2} - y^{2} + y^{4}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

$4 (1 - y^{2} - y^{2} + y^{4}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

$4 (1 - y^{2} - y^{2} + y^{4}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.3.2

Subtract $y^{2}$ from $- y^{2}$ .

$4 (1 - 2 y^{2} + y^{4}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

$4 (1 - 2 y^{2} + y^{4}) + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.4

Apply the distributive property.

$4 \cdot 1 + 4 (- 2 y^{2}) + 4 y^{4} + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.5

Simplify.

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Step 2.2.1.1.5.1

Multiply $4$ by $1$ .

$4 + 4 (- 2 y^{2}) + 4 y^{4} + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.1.5.2

Multiply $- 2$ by $4$ .

$4 - 8 y^{2} + 4 y^{4} + 9 y^{2} = 72$

$x = - 1 + y^{2}$

$4 - 8 y^{2} + 4 y^{4} + 9 y^{2} = 72$

$x = - 1 + y^{2}$

$4 - 8 y^{2} + 4 y^{4} + 9 y^{2} = 72$

$x = - 1 + y^{2}$

Step 2.2.1.2

Add $- 8 y^{2}$ and $9 y^{2}$ .

$4 + 4 y^{4} + y^{2} = 72$

$x = - 1 + y^{2}$

$4 + 4 y^{4} + y^{2} = 72$

$x = - 1 + y^{2}$

$4 + 4 y^{4} + y^{2} = 72$

$x = - 1 + y^{2}$

$4 + 4 y^{4} + y^{2} = 72$

$x = - 1 + y^{2}$

Step 3

Solve for $y$ in $4 + 4 y^{4} + y^{2} = 72$ .

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Step 3.1

Substitute $u = y^{2}$ into the equation. This will make the quadratic formula easy to use.

$4 u^{2} + u + 4 = 72$

$u = y^{2}$

$x = - 1 + y^{2}$

Step 3.2

Subtract $72$ from both sides of the equation.

$4 u^{2} + u + 4 - 72 = 0$

$x = - 1 + y^{2}$

Step 3.3

Subtract $72$ from $4$ .

$4 u^{2} + u - 68 = 0$

$x = - 1 + y^{2}$

Step 3.4

Factor by grouping.

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Step 3.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 68 = - 272$ and whose sum is $b = 1$ .

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Step 3.4.1.1

Multiply by $1$ .

$4 u^{2} + 1 u - 68 = 0$

$x = - 1 + y^{2}$

Step 3.4.1.2

Rewrite $1$ as $- 16$ plus $17$

$4 u^{2} + (- 16 + 17) u - 68 = 0$

$x = - 1 + y^{2}$

Step 3.4.1.3

Apply the distributive property.

$4 u^{2} - 16 u + 17 u - 68 = 0$

$x = - 1 + y^{2}$

$4 u^{2} - 16 u + 17 u - 68 = 0$

$x = - 1 + y^{2}$

Step 3.4.2

Factor out the greatest common factor from each group.

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Step 3.4.2.1

Group the first two terms and the last two terms.

$(4 u^{2} - 16 u) + 17 u - 68 = 0$

$x = - 1 + y^{2}$

Step 3.4.2.2

Factor out the greatest common factor (GCF) from each group.

$4 u (u - 4) + 17 (u - 4) = 0$

$x = - 1 + y^{2}$

$4 u (u - 4) + 17 (u - 4) = 0$

$x = - 1 + y^{2}$

Step 3.4.3

Factor the polynomial by factoring out the greatest common factor, $u - 4$ .

$(u - 4) (4 u + 17) = 0$

$x = - 1 + y^{2}$

$(u - 4) (4 u + 17) = 0$

$x = - 1 + y^{2}$

Step 3.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 4 = 0$

$4 u + 17 = 0$

$x = - 1 + y^{2}$

Step 3.6

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 3.6.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

$x = - 1 + y^{2}$

Step 3.6.2

Add $4$ to both sides of the equation.

$u = 4$

$x = - 1 + y^{2}$

$u = 4$

$x = - 1 + y^{2}$

Step 3.7

Set $4 u + 17$ equal to $0$ and solve for $u$ .

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Step 3.7.1

Set $4 u + 17$ equal to $0$ .

$4 u + 17 = 0$

$x = - 1 + y^{2}$

Step 3.7.2

Solve $4 u + 17 = 0$ for $u$ .

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Step 3.7.2.1

Subtract $17$ from both sides of the equation.

$4 u = - 17$

$x = - 1 + y^{2}$

Step 3.7.2.2

Divide each term in $4 u = - 17$ by $4$ and simplify.

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Step 3.7.2.2.1

Divide each term in $4 u = - 17$ by $4$ .

$\frac{4 u}{4} = \frac{- 17}{4}$

$x = - 1 + y^{2}$

Step 3.7.2.2.2

Simplify the left side.

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Step 3.7.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.7.2.2.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{- 17}{4}$

$x = - 1 + y^{2}$

Step 3.7.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 17}{4}$

$x = - 1 + y^{2}$

$u = \frac{- 17}{4}$

$x = - 1 + y^{2}$

$u = \frac{- 17}{4}$

$x = - 1 + y^{2}$

Step 3.7.2.2.3

Simplify the right side.

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Step 3.7.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{17}{4}$

$x = - 1 + y^{2}$

$u = - \frac{17}{4}$

$x = - 1 + y^{2}$

$u = - \frac{17}{4}$

$x = - 1 + y^{2}$

$u = - \frac{17}{4}$

$x = - 1 + y^{2}$

$u = - \frac{17}{4}$

$x = - 1 + y^{2}$

Step 3.8

The final solution is all the values that make $(u - 4) (4 u + 17) = 0$ true.

$u = 4, - \frac{17}{4}$

$x = - 1 + y^{2}$

Step 3.9

Substitute the real value of $u = y^{2}$ back into the solved equation.

$y^{2} = 4$

$y^{2} = - \frac{17}{4}$

$x = - 1 + y^{2}$

Step 3.10

Solve the first equation for $y$ .

$y^{2} = 4$

$x = - 1 + y^{2}$

Step 3.11

Solve the equation for $y$ .

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Step 3.11.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{4}$

$x = - 1 + y^{2}$

Step 3.11.2

Simplify $\pm \sqrt{4}$ .

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Step 3.11.2.1

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

$y = \pm \sqrt{2^{2}}$

$x = - 1 + y^{2}$

Step 3.11.2.2

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

$y = \pm 2$

$x = - 1 + y^{2}$

$y = \pm 2$

$x = - 1 + y^{2}$

Step 3.11.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.11.3.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 2$

$x = - 1 + y^{2}$

Step 3.11.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 2$

$x = - 1 + y^{2}$

Step 3.11.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = 2, - 2$

$x = - 1 + y^{2}$

$y = 2, - 2$

$x = - 1 + y^{2}$

$y = 2, - 2$

$x = - 1 + y^{2}$

Step 3.12

Solve the second equation for $y$ .

$y^{2} = - \frac{17}{4}$

$x = - 1 + y^{2}$

Step 3.13

Solve the equation for $y$ .

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Step 3.13.1

Remove parentheses.

$y^{2} = - \frac{17}{4}$

$x = - 1 + y^{2}$

Step 3.13.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- \frac{17}{4}}$

$x = - 1 + y^{2}$

Step 3.13.3

Simplify $\pm \sqrt{- \frac{17}{4}}$ .

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Step 3.13.3.1

Rewrite $- \frac{17}{4}$ as ${(\frac{i}{2})}^{2} \cdot 17$ .

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Step 3.13.3.1.1

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

$y = \pm \sqrt{i^{2} (\frac{17}{4})}$

$x = - 1 + y^{2}$

Step 3.13.3.1.2

Factor the perfect power $1^{2}$ out of $17$ .

$y = \pm \sqrt{i^{2} (\frac{1^{2} \cdot 17}{4})}$

$x = - 1 + y^{2}$

Step 3.13.3.1.3

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{i^{2} (\frac{1^{2} \cdot 17}{2^{2} \cdot 1})}$

$x = - 1 + y^{2}$

Step 3.13.3.1.4

Rearrange the fraction $\frac{1^{2} \cdot 17}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \cdot 17)}$

$x = - 1 + y^{2}$

Step 3.13.3.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$y = \pm \sqrt{{(\frac{i}{2})}^{2} \cdot 17}$

$x = - 1 + y^{2}$

$y = \pm \sqrt{{(\frac{i}{2})}^{2} \cdot 17}$

$x = - 1 + y^{2}$

Step 3.13.3.2

Pull terms out from under the radical.

$y = \pm \frac{i}{2} \cdot \sqrt{17}$

$x = - 1 + y^{2}$

Step 3.13.3.3

Combine $\frac{i}{2}$ and $\sqrt{17}$ .

$y = \pm \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

$y = \pm \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

Step 3.13.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.13.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

Step 3.13.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

Step 3.13.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{i \sqrt{17}}{2}, - \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

$y = \frac{i \sqrt{17}}{2}, - \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

$y = \frac{i \sqrt{17}}{2}, - \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

Step 3.14

The solution to $4 y^{4} + y^{2} + 4 = 72$ is $y = 2, - 2, \frac{i \sqrt{17}}{2}, - \frac{i \sqrt{17}}{2}$ .

$y = 2, - 2, \frac{i \sqrt{17}}{2}, - \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

$y = 2, - 2, \frac{i \sqrt{17}}{2}, - \frac{i \sqrt{17}}{2}$

$x = - 1 + y^{2}$

Step 4

Replace all occurrences of $y$ with $2$ in each equation.

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Step 4.1

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

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

$y = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 1 + {(2)}^{2}$ .

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Step 4.2.1.1

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

$x = - 1 + 4$

$y = 2$

Step 4.2.1.2

Add $- 1$ and $4$ .

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

Step 5

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 5.1

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

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

$y = - 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 1 + {(- 2)}^{2}$ .

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Step 5.2.1.1

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

$x = - 1 + 4$

$y = - 2$

Step 5.2.1.2

Add $- 1$ and $4$ .

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

Step 6

Replace all occurrences of $y$ with $2$ in each equation.

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Step 6.1

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

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

$y = 2$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 1 + {(2)}^{2}$ .

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Step 6.2.1.1

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

$x = - 1 + 4$

$y = 2$

Step 6.2.1.2

Add $- 1$ and $4$ .

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

Step 7

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 7.1

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

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

$y = - 2$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- 1 + {(- 2)}^{2}$ .

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Step 7.2.1.1

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

$x = - 1 + 4$

$y = - 2$

Step 7.2.1.2

Add $- 1$ and $4$ .

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

Step 8

Replace all occurrences of $y$ with $\frac{i \sqrt{17}}{2}$ in each equation.

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Step 8.1

Replace all occurrences of $y$ in $x = - 1 + y^{2}$ with $\frac{i \sqrt{17}}{2}$ .

$x = - 1 + {(\frac{i \sqrt{17}}{2})}^{2}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $- 1 + {(\frac{i \sqrt{17}}{2})}^{2}$ .

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

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

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Step 8.2.1.1.1.1

Apply the product rule to $\frac{i \sqrt{17}}{2}$ .

$x = - 1 + \frac{{(i \sqrt{17})}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.1.2

Apply the product rule to $i \sqrt{17}$ .

$x = - 1 + \frac{i^{2} {\sqrt{17}}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{i^{2} {\sqrt{17}}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.2

Simplify the numerator.

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Step 8.2.1.1.2.1

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

$x = - 1 + \frac{- 1 {\sqrt{17}}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.2.2

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 8.2.1.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$x = - 1 + \frac{- 1 {(17^{\frac{1}{2}})}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.2.2.2

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

$x = - 1 + \frac{- 1 \cdot 17^{\frac{1}{2} \cdot 2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.2.2.3

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

$x = - 1 + \frac{- 1 \cdot 17^{\frac{2}{2}}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.2.2.4

Cancel the common factor of $2$ .

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Step 8.2.1.1.2.2.4.1

Cancel the common factor.

$x = - 1 + \frac{- 1 \cdot 17^{\frac{2}{2}}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.2.2.4.2

Rewrite the expression.

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.2.2.5

Evaluate the exponent.

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.3

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

$x = - 1 + \frac{- 1 \cdot 17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.4

Multiply $- 1$ by $17$ .

$x = - 1 + \frac{- 17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.1.5

Move the negative in front of the fraction.

$x = - 1 - \frac{17}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 - \frac{17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - 1 \cdot \frac{4}{4} - \frac{17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.3

Combine $- 1$ and $\frac{4}{4}$ .

$x = \frac{- 1 \cdot 4}{4} - \frac{17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 1 \cdot 4 - 17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.5

Simplify the numerator.

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Step 8.2.1.5.1

Multiply $- 1$ by $4$ .

$x = \frac{- 4 - 17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.5.2

Subtract $17$ from $- 4$ .

$x = \frac{- 21}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = \frac{- 21}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 8.2.1.6

Move the negative in front of the fraction.

$x = - \frac{21}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 9

Replace all occurrences of $y$ with $2$ in each equation.

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Step 9.1

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

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

$y = 2$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $- 1 + {(2)}^{2}$ .

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Step 9.2.1.1

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

$x = - 1 + 4$

$y = 2$

Step 9.2.1.2

Add $- 1$ and $4$ .

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

$x = 3$

$y = 2$

Step 10

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 10.1

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

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

$y = - 2$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $- 1 + {(- 2)}^{2}$ .

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Step 10.2.1.1

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

$x = - 1 + 4$

$y = - 2$

Step 10.2.1.2

Add $- 1$ and $4$ .

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

$x = 3$

$y = - 2$

Step 11

Replace all occurrences of $y$ with $\frac{i \sqrt{17}}{2}$ in each equation.

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Step 11.1

Replace all occurrences of $y$ in $x = - 1 + y^{2}$ with $\frac{i \sqrt{17}}{2}$ .

$x = - 1 + {(\frac{i \sqrt{17}}{2})}^{2}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $- 1 + {(\frac{i \sqrt{17}}{2})}^{2}$ .

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Step 11.2.1.1

Simplify each term.

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Step 11.2.1.1.1

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

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Step 11.2.1.1.1.1

Apply the product rule to $\frac{i \sqrt{17}}{2}$ .

$x = - 1 + \frac{{(i \sqrt{17})}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.1.2

Apply the product rule to $i \sqrt{17}$ .

$x = - 1 + \frac{i^{2} {\sqrt{17}}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{i^{2} {\sqrt{17}}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.2

Simplify the numerator.

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Step 11.2.1.1.2.1

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

$x = - 1 + \frac{- 1 {\sqrt{17}}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.2.2

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 11.2.1.1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$x = - 1 + \frac{- 1 {(17^{\frac{1}{2}})}^{2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.2.2.2

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

$x = - 1 + \frac{- 1 \cdot 17^{\frac{1}{2} \cdot 2}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.2.2.3

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

$x = - 1 + \frac{- 1 \cdot 17^{\frac{2}{2}}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.2.2.4

Cancel the common factor of $2$ .

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Step 11.2.1.1.2.2.4.1

Cancel the common factor.

$x = - 1 + \frac{- 1 \cdot 17^{\frac{2}{2}}}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.2.2.4.2

Rewrite the expression.

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.2.2.5

Evaluate the exponent.

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.3

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

$x = - 1 + \frac{- 1 \cdot 17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.4

Multiply $- 1$ by $17$ .

$x = - 1 + \frac{- 17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.1.5

Move the negative in front of the fraction.

$x = - 1 - \frac{17}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = - 1 - \frac{17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - 1 \cdot \frac{4}{4} - \frac{17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.3

Combine $- 1$ and $\frac{4}{4}$ .

$x = \frac{- 1 \cdot 4}{4} - \frac{17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 1 \cdot 4 - 17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.5

Simplify the numerator.

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Step 11.2.1.5.1

Multiply $- 1$ by $4$ .

$x = \frac{- 4 - 17}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.5.2

Subtract $17$ from $- 4$ .

$x = \frac{- 21}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = \frac{- 21}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 11.2.1.6

Move the negative in front of the fraction.

$x = - \frac{21}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = \frac{i \sqrt{17}}{2}$

Step 12

Replace all occurrences of $y$ with $- \frac{i \sqrt{17}}{2}$ in each equation.

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Step 12.1

Replace all occurrences of $y$ in $x = - 1 + y^{2}$ with $- \frac{i \sqrt{17}}{2}$ .

$x = - 1 + {(- \frac{i \sqrt{17}}{2})}^{2}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $- 1 + {(- \frac{i \sqrt{17}}{2})}^{2}$ .

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Step 12.2.1.1

Simplify each term.

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Step 12.2.1.1.1

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

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Step 12.2.1.1.1.1

Apply the product rule to $- \frac{i \sqrt{17}}{2}$ .

$x = - 1 + {(- 1)}^{2} {(\frac{i \sqrt{17}}{2})}^{2}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.1.2

Apply the product rule to $\frac{i \sqrt{17}}{2}$ .

$x = - 1 + {(- 1)}^{2} (\frac{{(i \sqrt{17})}^{2}}{2^{2}})$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.1.3

Apply the product rule to $i \sqrt{17}$ .

$x = - 1 + {(- 1)}^{2} (\frac{i^{2} {\sqrt{17}}^{2}}{2^{2}})$

$y = - \frac{i \sqrt{17}}{2}$

$x = - 1 + {(- 1)}^{2} (\frac{i^{2} {\sqrt{17}}^{2}}{2^{2}})$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.2

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

$x = - 1 + 1 (\frac{i^{2} {\sqrt{17}}^{2}}{2^{2}})$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.3

Multiply $\frac{i^{2} {\sqrt{17}}^{2}}{2^{2}}$ by $1$ .

$x = - 1 + \frac{i^{2} {\sqrt{17}}^{2}}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.4

Simplify the numerator.

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Step 12.2.1.1.4.1

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

$x = - 1 + \frac{- 1 {\sqrt{17}}^{2}}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.4.2

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 12.2.1.1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$x = - 1 + \frac{- 1 {(17^{\frac{1}{2}})}^{2}}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.4.2.2

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

$x = - 1 + \frac{- 1 \cdot 17^{\frac{1}{2} \cdot 2}}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.4.2.3

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

$x = - 1 + \frac{- 1 \cdot 17^{\frac{2}{2}}}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.4.2.4

Cancel the common factor of $2$ .

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Step 12.2.1.1.4.2.4.1

Cancel the common factor.

$x = - 1 + \frac{- 1 \cdot 17^{\frac{2}{2}}}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.4.2.4.2

Rewrite the expression.

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.4.2.5

Evaluate the exponent.

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

$x = - 1 + \frac{- 1 \cdot 17}{2^{2}}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.5

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

$x = - 1 + \frac{- 1 \cdot 17}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.6

Multiply $- 1$ by $17$ .

$x = - 1 + \frac{- 17}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.1.7

Move the negative in front of the fraction.

$x = - 1 - \frac{17}{4}$

$y = - \frac{i \sqrt{17}}{2}$

$x = - 1 - \frac{17}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - 1 \cdot \frac{4}{4} - \frac{17}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.3

Combine $- 1$ and $\frac{4}{4}$ .

$x = \frac{- 1 \cdot 4}{4} - \frac{17}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 1 \cdot 4 - 17}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.5

Simplify the numerator.

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Step 12.2.1.5.1

Multiply $- 1$ by $4$ .

$x = \frac{- 4 - 17}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.5.2

Subtract $17$ from $- 4$ .

$x = \frac{- 21}{4}$

$y = - \frac{i \sqrt{17}}{2}$

$x = \frac{- 21}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 12.2.1.6

Move the negative in front of the fraction.

$x = - \frac{21}{4}$

$y = - \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = - \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = - \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}$

$y = - \frac{i \sqrt{17}}{2}$

Step 13

List all of the solutions.

$x = 3, y = 2$

$x = 3, y = - 2$

$x = - \frac{21}{4}, y = \frac{i \sqrt{17}}{2}$

$x = - \frac{21}{4}, y = - \frac{i \sqrt{17}}{2}$

Step 14