Solve the System of Equations y=x^2-2x+2 x+y=1

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

Step 1

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

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

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

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

Simplify the left side.

Simplify $x + x^{2} - 2 x + 2$ .

Remove parentheses.

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

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

Subtract $2 x$ from $x$ .

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

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

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

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

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

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

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

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

Step 2

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

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

Subtract $1$ from both sides of the equation.

$x^{2} - x + 2 - 1 = 0$

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

Subtract $1$ from $2$ .

$x^{2} - x + 1 = 0$

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

$x^{2} - x + 1 = 0$

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

Use the quadratic formula to find the solutions.

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

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

Substitute the values $a = 1$ , $b = - 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

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

Simplify.

Simplify the numerator.

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

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

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

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

Subtract $4$ from $1$ .

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

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

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

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

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

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

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

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

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

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

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

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

$x = \frac{1 \pm i \sqrt{3}}{2}$

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

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

Simplify the numerator.

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

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

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

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

Subtract $4$ from $1$ .

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

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

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

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

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

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

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

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

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

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

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

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

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{3}}{2}$

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

$x = \frac{1 + i \sqrt{3}}{2}$

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

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

Simplify the numerator.

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4 \cdot 1 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

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

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

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

Subtract $4$ from $1$ .

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

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

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

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

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

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

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

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

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

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

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

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

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

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

Step 3

Replace all occurrences of $x$ with $\frac{1 + i \sqrt{3}}{2}$ in each equation.

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

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

$x = \frac{1 + i \sqrt{3}}{2}$

Simplify the right side.

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

Simplify each term.

Apply the product rule to $\frac{1 + i \sqrt{3}}{2}$ .

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

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{{(1 + i \sqrt{3})}^{2}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{(1 + i \sqrt{3}) (1 + i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Expand $(1 + i \sqrt{3}) (1 + i \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{1 (1 + i \sqrt{3}) + i \sqrt{3} (1 + i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (i \sqrt{3}) + i \sqrt{3} (1 + i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (i \sqrt{3}) + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 \cdot 1 + 1 (i \sqrt{3}) + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$y = \frac{1 + 1 (i \sqrt{3}) + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Multiply $i \sqrt{3}$ by $1$ .

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} \cdot 1 + i \sqrt{3} (i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Multiply $i$ by $1$ .

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i \sqrt{3} (i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Multiply $i \sqrt{3} (i \sqrt{3})$ .

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

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i i \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i i \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{1 + 1} \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Add $1$ and $1$ .

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} (\sqrt{3} \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} (\sqrt{3} \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} {\sqrt{3}}^{1 + 1}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Add $1$ and $1$ .

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} {\sqrt{3}}^{2}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} + i^{2} {\sqrt{3}}^{2}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 {\sqrt{3}}^{2}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

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

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 {(3^{\frac{1}{2}})}^{2}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{1}{2} \cdot 2}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Rewrite the expression.

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Evaluate the exponent.

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 1 \cdot 3}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Multiply $- 1$ by $3$ .

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 3}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 + i \sqrt{3} + i \sqrt{3} - 3}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Subtract $3$ from $1$ .

$y = \frac{i \sqrt{3} + i \sqrt{3} - 2}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Add $i \sqrt{3}$ and $i \sqrt{3}$ .

$y = \frac{2 i \sqrt{3} - 2}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{2 i \sqrt{3} - 2}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Reorder $2 i \sqrt{3}$ and $- 2$ .

$y = \frac{- 2 + 2 i \sqrt{3}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Cancel the common factor of $- 2 + 2 i \sqrt{3}$ and $4$ .

Factor $2$ out of $- 2$ .

$y = \frac{2 \cdot - 1 + 2 i \sqrt{3}}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Factor $2$ out of $2 i \sqrt{3}$ .

$y = \frac{2 \cdot - 1 + 2 (i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Factor $2$ out of $2 \cdot - 1 + 2 (i \sqrt{3})$ .

$y = \frac{2 \cdot (- 1 + i \sqrt{3})}{4} - 2 \frac{1 + i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

$x = \frac{1 + i \sqrt{3}}{2}$

Cancel the common factor.

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

$x = \frac{1 + i \sqrt{3}}{2}$

Rewrite the expression.

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

$x = \frac{1 + i \sqrt{3}}{2}$

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

$x = \frac{1 + i \sqrt{3}}{2}$

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

$x = \frac{1 + i \sqrt{3}}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

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

$x = \frac{1 + i \sqrt{3}}{2}$

Cancel the common factor.

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

$x = \frac{1 + i \sqrt{3}}{2}$

Rewrite the expression.

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

$x = \frac{1 + i \sqrt{3}}{2}$

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

$x = \frac{1 + i \sqrt{3}}{2}$

Apply the distributive property.

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

$x = \frac{1 + i \sqrt{3}}{2}$

Multiply $- 1$ by $1$ .

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

$x = \frac{1 + i \sqrt{3}}{2}$

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

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

$x = \frac{1 + i \sqrt{3}}{2}$

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

$x = \frac{1 + i \sqrt{3}}{2}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

$x = \frac{1 + i \sqrt{3}}{2}$

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

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

$x = \frac{1 + i \sqrt{3}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 3 + i \sqrt{3}}{2} - i \sqrt{3} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{- 3 + i \sqrt{3}}{2} - i \sqrt{3} \cdot \frac{2}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Combine $- i \sqrt{3}$ and $\frac{2}{2}$ .

$y = \frac{- 3 + i \sqrt{3}}{2} + \frac{- i \sqrt{3} \cdot 2}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 3 - i \sqrt{3}}{2} + 2$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{- 3 - i \sqrt{3}}{2} + 2 \cdot \frac{2}{2}$

$x = \frac{1 + i \sqrt{3}}{2}$

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

$y = \frac{- 3 - i \sqrt{3}}{2} + \frac{2 \cdot 2}{2}$

$x = \frac{1 + i \sqrt{3}}{2}$

Combine the numerators over the common denominator.

$y = \frac{1 - i \sqrt{3}}{2}$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 - i \sqrt{3}}{2}$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 - i \sqrt{3}}{2}$

$x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 - i \sqrt{3}}{2}$

$x = \frac{1 + i \sqrt{3}}{2}$

Step 4

Replace all occurrences of $x$ with $\frac{1 - i \sqrt{3}}{2}$ in each equation.

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

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

$x = \frac{1 - i \sqrt{3}}{2}$

Simplify the right side.

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

Simplify each term.

Apply the product rule to $\frac{1 - i \sqrt{3}}{2}$ .

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{{(1 - i \sqrt{3})}^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Rewrite ${(1 - i \sqrt{3})}^{2}$ as $(1 - i \sqrt{3}) (1 - i \sqrt{3})$ .

$y = \frac{(1 - i \sqrt{3}) (1 - i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Expand $(1 - i \sqrt{3}) (1 - i \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{1 (1 - i \sqrt{3}) - i \sqrt{3} (1 - i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (- i \sqrt{3}) - i \sqrt{3} (1 - i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Apply the distributive property.

$y = \frac{1 \cdot 1 + 1 (- i \sqrt{3}) - i \sqrt{3} \cdot 1 - i \sqrt{3} (- i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

$y = \frac{1 \cdot 1 + 1 (- i \sqrt{3}) - i \sqrt{3} \cdot 1 - i \sqrt{3} (- i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$y = \frac{1 + 1 (- i \sqrt{3}) - i \sqrt{3} \cdot 1 - i \sqrt{3} (- i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Multiply $- i \sqrt{3}$ by $1$ .

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} \cdot 1 - i \sqrt{3} (- i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Multiply $- 1$ by $1$ .

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} - i \sqrt{3} (- i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Multiply $- i \sqrt{3} (- i \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 1 i \sqrt{3} (i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + \sqrt{3} i (i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + \sqrt{3} \sqrt{3} i i}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + \sqrt{3} \sqrt{3} i i}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{1 + 1} i i}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Add $1$ and $1$ .

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} i i}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} (i i)}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} (i i)}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} i^{1 + 1}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Add $1$ and $1$ .

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + {\sqrt{3}}^{2} i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + {(3^{\frac{1}{2}})}^{2} i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3^{\frac{1}{2} \cdot 2} i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3^{\frac{2}{2}} i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3^{\frac{2}{2}} i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Rewrite the expression.

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3 i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3 i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Evaluate the exponent.

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3 i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3 i^{2}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} + 3 \cdot - 1}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Multiply $3$ by $- 1$ .

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} - 3}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

$y = \frac{1 - i \sqrt{3} - i \sqrt{3} - 3}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Subtract $3$ from $1$ .

$y = \frac{- i \sqrt{3} - i \sqrt{3} - 2}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Subtract $i \sqrt{3}$ from $- i \sqrt{3}$ .

$y = \frac{- 2 i \sqrt{3} - 2}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

$y = \frac{- 2 i \sqrt{3} - 2}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Reorder $- 2 i \sqrt{3}$ and $- 2$ .

$y = \frac{- 2 - 2 i \sqrt{3}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Cancel the common factor of $- 2 - 2 i \sqrt{3}$ and $4$ .

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1) - 2 i \sqrt{3}}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Factor $2$ out of $- 2 i \sqrt{3}$ .

$y = \frac{2 (- 1) + 2 (- i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Factor $2$ out of $2 (- 1) + 2 (- i \sqrt{3})$ .

$y = \frac{2 (- 1 - i \sqrt{3})}{4} - 2 \frac{1 - i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

$x = \frac{1 - i \sqrt{3}}{2}$

Cancel the common factor.

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

$x = \frac{1 - i \sqrt{3}}{2}$

Rewrite the expression.

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

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

$x = \frac{1 - i \sqrt{3}}{2}$

Cancel the common factor.

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

$x = \frac{1 - i \sqrt{3}}{2}$

Rewrite the expression.

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

Apply the distributive property.

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

$x = \frac{1 - i \sqrt{3}}{2}$

Multiply $- 1$ by $1$ .

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

$x = \frac{1 - i \sqrt{3}}{2}$

Multiply $- 1 (- i \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

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

$x = \frac{1 - i \sqrt{3}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 3 - i \sqrt{3}}{2} + \sqrt{3} i + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{- 3 - i \sqrt{3}}{2} + \sqrt{3} i \cdot \frac{2}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Combine $\sqrt{3} i$ and $\frac{2}{2}$ .

$y = \frac{- 3 - i \sqrt{3}}{2} + \frac{\sqrt{3} i \cdot 2}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 3 + i \sqrt{3}}{2} + 2$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{- 3 + i \sqrt{3}}{2} + 2 \cdot \frac{2}{2}$

$x = \frac{1 - i \sqrt{3}}{2}$

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

$y = \frac{- 3 + i \sqrt{3}}{2} + \frac{2 \cdot 2}{2}$

$x = \frac{1 - i \sqrt{3}}{2}$

Combine the numerators over the common denominator.

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

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

$x = \frac{1 - i \sqrt{3}}{2}$

Step 5

List all of the solutions.

$y = \frac{1 - i \sqrt{3}}{2}, x = \frac{1 + i \sqrt{3}}{2}$

$y = \frac{1 + i \sqrt{3}}{2}, x = \frac{1 - i \sqrt{3}}{2}$

Step 6

Solve the System of Equations y=x^2-2x+2 x+y=1

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