Algebra Examples

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

Step 1

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

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Step 1.1

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

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

$y = x^{2} + 1$

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

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

$y = x^{2} + 1$

Step 1.2.1.1.2

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

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Step 1.2.1.1.2.1

Apply the distributive property.

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

$y = x^{2} + 1$

Step 1.2.1.1.2.2

Apply the distributive property.

$x^{2} + x^{2} x^{2} + x^{2} \cdot 1 + 1 (x^{2} + 1) = 1$

$y = x^{2} + 1$

Step 1.2.1.1.2.3

Apply the distributive property.

$x^{2} + x^{2} x^{2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1 = 1$

$y = x^{2} + 1$

$x^{2} + x^{2} x^{2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1 = 1$

$y = x^{2} + 1$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

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

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Step 1.2.1.1.3.1.1.1

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

$x^{2} + x^{2 + 2} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1 = 1$

$y = x^{2} + 1$

Step 1.2.1.1.3.1.1.2

Add $2$ and $2$ .

$x^{2} + x^{4} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1 = 1$

$y = x^{2} + 1$

$x^{2} + x^{4} + x^{2} \cdot 1 + 1 x^{2} + 1 \cdot 1 = 1$

$y = x^{2} + 1$

Step 1.2.1.1.3.1.2

Multiply $x^{2}$ by $1$ .

$x^{2} + x^{4} + x^{2} + 1 x^{2} + 1 \cdot 1 = 1$

$y = x^{2} + 1$

Step 1.2.1.1.3.1.3

Multiply $x^{2}$ by $1$ .

$x^{2} + x^{4} + x^{2} + x^{2} + 1 \cdot 1 = 1$

$y = x^{2} + 1$

Step 1.2.1.1.3.1.4

Multiply $1$ by $1$ .

$x^{2} + x^{4} + x^{2} + x^{2} + 1 = 1$

$y = x^{2} + 1$

$x^{2} + x^{4} + x^{2} + x^{2} + 1 = 1$

$y = x^{2} + 1$

Step 1.2.1.1.3.2

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

$x^{2} + x^{4} + 2 x^{2} + 1 = 1$

$y = x^{2} + 1$

$x^{2} + x^{4} + 2 x^{2} + 1 = 1$

$y = x^{2} + 1$

$x^{2} + x^{4} + 2 x^{2} + 1 = 1$

$y = x^{2} + 1$

Step 1.2.1.2

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

$x^{4} + 3 x^{2} + 1 = 1$

$y = x^{2} + 1$

$x^{4} + 3 x^{2} + 1 = 1$

$y = x^{2} + 1$

$x^{4} + 3 x^{2} + 1 = 1$

$y = x^{2} + 1$

$x^{4} + 3 x^{2} + 1 = 1$

$y = x^{2} + 1$

Step 2

Solve for $x$ in $x^{4} + 3 x^{2} + 1 = 1$ .

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Step 2.1

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

$u^{2} + 3 u + 1 = 1$

$u = x^{2}$

$y = x^{2} + 1$

Step 2.2

Subtract $1$ from both sides of the equation.

$u^{2} + 3 u + 1 - 1 = 0$

$y = x^{2} + 1$

Step 2.3

Combine the opposite terms in $u^{2} + 3 u + 1 - 1$ .

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Step 2.3.1

Subtract $1$ from $1$ .

$u^{2} + 3 u + 0 = 0$

$y = x^{2} + 1$

Step 2.3.2

Add $u^{2} + 3 u$ and $0$ .

$u^{2} + 3 u = 0$

$y = x^{2} + 1$

$u^{2} + 3 u = 0$

$y = x^{2} + 1$

Step 2.4

Factor $u$ out of $u^{2} + 3 u$ .

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Step 2.4.1

Factor $u$ out of $u^{2}$ .

$u \cdot u + 3 u = 0$

$y = x^{2} + 1$

Step 2.4.2

Factor $u$ out of $3 u$ .

$u \cdot u + u \cdot 3 = 0$

$y = x^{2} + 1$

Step 2.4.3

Factor $u$ out of $u \cdot u + u \cdot 3$ .

$u (u + 3) = 0$

$y = x^{2} + 1$

$u (u + 3) = 0$

$y = x^{2} + 1$

Step 2.5

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

$u = 0$

$u + 3 = 0$

$y = x^{2} + 1$

Step 2.6

Set $u$ equal to $0$ .

$u = 0$

$y = x^{2} + 1$

Step 2.7

Set $u + 3$ equal to $0$ and solve for $u$ .

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Step 2.7.1

Set $u + 3$ equal to $0$ .

$u + 3 = 0$

$y = x^{2} + 1$

Step 2.7.2

Subtract $3$ from both sides of the equation.

$u = - 3$

$y = x^{2} + 1$

$u = - 3$

$y = x^{2} + 1$

Step 2.8

The final solution is all the values that make $u (u + 3) = 0$ true.

$u = 0, - 3$

$y = x^{2} + 1$

Step 2.9

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

$x^{2} = 0$

$x^{2} = - 3$

$y = x^{2} + 1$

Step 2.10

Solve the first equation for $x$ .

$x^{2} = 0$

$y = x^{2} + 1$

Step 2.11

Solve the equation for $x$ .

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Step 2.11.1

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

$x = \pm \sqrt{0}$

$y = x^{2} + 1$

Step 2.11.2

Simplify $\pm \sqrt{0}$ .

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Step 2.11.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

$y = x^{2} + 1$

Step 2.11.2.2

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

$x = \pm 0$

$y = x^{2} + 1$

Step 2.11.2.3

Plus or minus $0$ is $0$ .

$x = 0$

$y = x^{2} + 1$

$x = 0$

$y = x^{2} + 1$

$x = 0$

$y = x^{2} + 1$

Step 2.12

Solve the second equation for $x$ .

$x^{2} = - 3$

$y = x^{2} + 1$

Step 2.13

Solve the equation for $x$ .

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Step 2.13.1

Remove parentheses.

$x^{2} = - 3$

$y = x^{2} + 1$

Step 2.13.2

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

$x = \pm \sqrt{- 3}$

$y = x^{2} + 1$

Step 2.13.3

Simplify $\pm \sqrt{- 3}$ .

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Step 2.13.3.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 \cdot 3}$

$y = x^{2} + 1$

Step 2.13.3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

$y = x^{2} + 1$

Step 2.13.3.3

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

$x = \pm i \sqrt{3}$

$y = x^{2} + 1$

$x = \pm i \sqrt{3}$

$y = x^{2} + 1$

Step 2.13.4

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

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Step 2.13.4.1

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

$x = i \sqrt{3}$

$y = x^{2} + 1$

Step 2.13.4.2

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

$x = - i \sqrt{3}$

$y = x^{2} + 1$

Step 2.13.4.3

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

$x = i \sqrt{3}, - i \sqrt{3}$

$y = x^{2} + 1$

$x = i \sqrt{3}, - i \sqrt{3}$

$y = x^{2} + 1$

$x = i \sqrt{3}, - i \sqrt{3}$

$y = x^{2} + 1$

Step 2.14

The solution to $x^{4} + 3 x^{2} + 1 = 1$ is $x = 0, i \sqrt{3}, - i \sqrt{3}$ .

$x = 0, i \sqrt{3}, - i \sqrt{3}$

$y = x^{2} + 1$

$x = 0, i \sqrt{3}, - i \sqrt{3}$

$y = x^{2} + 1$

Step 3

Replace all occurrences of $x$ with $0$ in each equation.

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Step 3.1

Replace all occurrences of $x$ in $y = x^{2} + 1$ with $0$ .

$y = {(0)}^{2} + 1$

$x = 0$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(0)}^{2} + 1$ .

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Step 3.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 + 1$

$x = 0$

Step 3.2.1.2

Add $0$ and $1$ .

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

Step 4

Replace all occurrences of $x$ with $i \sqrt{3}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = x^{2} + 1$ with $i \sqrt{3}$ .

$y = {(i \sqrt{3})}^{2} + 1$

$x = i \sqrt{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(i \sqrt{3})}^{2} + 1$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the product rule to $i \sqrt{3}$ .

$y = i^{2} {\sqrt{3}}^{2} + 1$

$x = i \sqrt{3}$

Step 4.2.1.1.2

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

$y = - 1 {\sqrt{3}}^{2} + 1$

$x = i \sqrt{3}$

Step 4.2.1.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

$x = i \sqrt{3}$

Step 4.2.1.1.3.2

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

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

$x = i \sqrt{3}$

Step 4.2.1.1.3.3

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

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

$x = i \sqrt{3}$

Step 4.2.1.1.3.4

Cancel the common factor of $2$ .

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Step 4.2.1.1.3.4.1

Cancel the common factor.

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

$x = i \sqrt{3}$

Step 4.2.1.1.3.4.2

Rewrite the expression.

$y = - 1 \cdot 3 + 1$

$x = i \sqrt{3}$

$y = - 1 \cdot 3 + 1$

$x = i \sqrt{3}$

Step 4.2.1.1.3.5

Evaluate the exponent.

$y = - 1 \cdot 3 + 1$

$x = i \sqrt{3}$

$y = - 1 \cdot 3 + 1$

$x = i \sqrt{3}$

Step 4.2.1.1.4

Multiply $- 1$ by $3$ .

$y = - 3 + 1$

$x = i \sqrt{3}$

$y = - 3 + 1$

$x = i \sqrt{3}$

Step 4.2.1.2

Add $- 3$ and $1$ .

$y = - 2$

$x = i \sqrt{3}$

$y = - 2$

$x = i \sqrt{3}$

$y = - 2$

$x = i \sqrt{3}$

$y = - 2$

$x = i \sqrt{3}$

Step 5

Replace all occurrences of $x$ with $0$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = x^{2} + 1$ with $0$ .

$y = {(0)}^{2} + 1$

$x = 0$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify ${(0)}^{2} + 1$ .

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Step 5.2.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 + 1$

$x = 0$

Step 5.2.1.2

Add $0$ and $1$ .

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

$y = 1$

$x = 0$

Step 6

Replace all occurrences of $x$ with $i \sqrt{3}$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = x^{2} + 1$ with $i \sqrt{3}$ .

$y = {(i \sqrt{3})}^{2} + 1$

$x = i \sqrt{3}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify ${(i \sqrt{3})}^{2} + 1$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Apply the product rule to $i \sqrt{3}$ .

$y = i^{2} {\sqrt{3}}^{2} + 1$

$x = i \sqrt{3}$

Step 6.2.1.1.2

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

$y = - 1 {\sqrt{3}}^{2} + 1$

$x = i \sqrt{3}$

Step 6.2.1.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.2.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

$x = i \sqrt{3}$

Step 6.2.1.1.3.2

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

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

$x = i \sqrt{3}$

Step 6.2.1.1.3.3

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

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

$x = i \sqrt{3}$

Step 6.2.1.1.3.4

Cancel the common factor of $2$ .

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Step 6.2.1.1.3.4.1

Cancel the common factor.

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

$x = i \sqrt{3}$

Step 6.2.1.1.3.4.2

Rewrite the expression.

$y = - 1 \cdot 3 + 1$

$x = i \sqrt{3}$

$y = - 1 \cdot 3 + 1$

$x = i \sqrt{3}$

Step 6.2.1.1.3.5

Evaluate the exponent.

$y = - 1 \cdot 3 + 1$

$x = i \sqrt{3}$

$y = - 1 \cdot 3 + 1$

$x = i \sqrt{3}$

Step 6.2.1.1.4

Multiply $- 1$ by $3$ .

$y = - 3 + 1$

$x = i \sqrt{3}$

$y = - 3 + 1$

$x = i \sqrt{3}$

Step 6.2.1.2

Add $- 3$ and $1$ .

$y = - 2$

$x = i \sqrt{3}$

$y = - 2$

$x = i \sqrt{3}$

$y = - 2$

$x = i \sqrt{3}$

$y = - 2$

$x = i \sqrt{3}$

Step 7

Replace all occurrences of $x$ with $- i \sqrt{3}$ in each equation.

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Step 7.1

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

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

$x = - i \sqrt{3}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify ${(- i \sqrt{3})}^{2} + 1$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

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

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Step 7.2.1.1.1.1

Apply the product rule to $- i \sqrt{3}$ .

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

$x = - i \sqrt{3}$

Step 7.2.1.1.1.2

Apply the product rule to $- i$ .

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

$x = - i \sqrt{3}$

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

$x = - i \sqrt{3}$

Step 7.2.1.1.2

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

$y = 1 i^{2} {\sqrt{3}}^{2} + 1$

$x = - i \sqrt{3}$

Step 7.2.1.1.3

Multiply $i^{2}$ by $1$ .

$y = i^{2} {\sqrt{3}}^{2} + 1$

$x = - i \sqrt{3}$

Step 7.2.1.1.4

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

$y = - 1 {\sqrt{3}}^{2} + 1$

$x = - i \sqrt{3}$

Step 7.2.1.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.2.1.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

$x = - i \sqrt{3}$

Step 7.2.1.1.5.2

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

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

$x = - i \sqrt{3}$

Step 7.2.1.1.5.3

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

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

$x = - i \sqrt{3}$

Step 7.2.1.1.5.4

Cancel the common factor of $2$ .

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Step 7.2.1.1.5.4.1

Cancel the common factor.

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

$x = - i \sqrt{3}$

Step 7.2.1.1.5.4.2

Rewrite the expression.

$y = - 1 \cdot 3 + 1$

$x = - i \sqrt{3}$

$y = - 1 \cdot 3 + 1$

$x = - i \sqrt{3}$

Step 7.2.1.1.5.5

Evaluate the exponent.

$y = - 1 \cdot 3 + 1$

$x = - i \sqrt{3}$

$y = - 1 \cdot 3 + 1$

$x = - i \sqrt{3}$

Step 7.2.1.1.6

Multiply $- 1$ by $3$ .

$y = - 3 + 1$

$x = - i \sqrt{3}$

$y = - 3 + 1$

$x = - i \sqrt{3}$

Step 7.2.1.2

Add $- 3$ and $1$ .

$y = - 2$

$x = - i \sqrt{3}$

$y = - 2$

$x = - i \sqrt{3}$

$y = - 2$

$x = - i \sqrt{3}$

$y = - 2$

$x = - i \sqrt{3}$

Step 8

List all of the solutions.

$y = 1, x = 0$

$y = - 2, x = i \sqrt{3}$

$y = - 2, x = - i \sqrt{3}$

Step 9