Solve the System of Equations y-6=x^2-3x x+2y=2

$y - 6 = x^{2} - 3 x$ $x + 2 y = 2$

Add $6$ to both sides of the equation.

$y = x^{2} - 3 x + 6$

$x + 2 y = 2$

Replace all occurrences of $y$ with $x^{2} - 3 x + 6$ in each equation.

Replace all occurrences of $y$ in $x + 2 y = 2$ with $x^{2} - 3 x + 6$ .

$x + 2 (x^{2} - 3 x + 6) = 2$

$y = x^{2} - 3 x + 6$

Simplify the left side.

Simplify $x + 2 (x^{2} - 3 x + 6)$ .

Simplify each term.

Apply the distributive property.

$x + 2 x^{2} + 2 (- 3 x) + 2 \cdot 6 = 2$

$y = x^{2} - 3 x + 6$

Simplify.

Multiply $- 3$ by $2$ .

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

$y = x^{2} - 3 x + 6$

Multiply $2$ by $6$ .

$x + 2 x^{2} - 6 x + 12 = 2$

$y = x^{2} - 3 x + 6$

$x + 2 x^{2} - 6 x + 12 = 2$

$y = x^{2} - 3 x + 6$

$x + 2 x^{2} - 6 x + 12 = 2$

$y = x^{2} - 3 x + 6$

Subtract $6 x$ from $x$ .

$2 x^{2} - 5 x + 12 = 2$

$y = x^{2} - 3 x + 6$

$2 x^{2} - 5 x + 12 = 2$

$y = x^{2} - 3 x + 6$

$2 x^{2} - 5 x + 12 = 2$

$y = x^{2} - 3 x + 6$

$2 x^{2} - 5 x + 12 = 2$

$y = x^{2} - 3 x + 6$

Solve for $x$ in $2 x^{2} - 5 x + 12 = 2$ .

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

Subtract $2$ from both sides of the equation.

$2 x^{2} - 5 x + 12 - 2 = 0$

$y = x^{2} - 3 x + 6$

Subtract $2$ from $12$ .

$2 x^{2} - 5 x + 10 = 0$

$y = x^{2} - 3 x + 6$

$2 x^{2} - 5 x + 10 = 0$

$y = x^{2} - 3 x + 6$

Use the quadratic formula to find the solutions.

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

$y = x^{2} - 3 x + 6$

Substitute the values $a = 2$ , $b = - 5$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (2 \cdot 10)}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Simplify.

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 2 \cdot 10}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot 10}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $- 8$ by $10$ .

$x = \frac{5 \pm \sqrt{25 - 80}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Subtract $80$ from $25$ .

$x = \frac{5 \pm \sqrt{- 55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Rewrite $- 55$ as $- 1 (55)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Rewrite $\sqrt{- 1 (55)}$ as $\sqrt{- 1} \cdot \sqrt{55}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

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

$x = \frac{5 \pm i \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

$x = \frac{5 \pm i \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $2$ by $2$ .

$x = \frac{5 \pm i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

$x = \frac{5 \pm i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

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

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 2 \cdot 10}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot 10}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $- 8$ by $10$ .

$x = \frac{5 \pm \sqrt{25 - 80}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Subtract $80$ from $25$ .

$x = \frac{5 \pm \sqrt{- 55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Rewrite $- 55$ as $- 1 (55)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Rewrite $\sqrt{- 1 (55)}$ as $\sqrt{- 1} \cdot \sqrt{55}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

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

$x = \frac{5 \pm i \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

$x = \frac{5 \pm i \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $2$ by $2$ .

$x = \frac{5 \pm i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

Change the $\pm$ to $+$ .

$x = \frac{5 + i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

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

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 2 \cdot 10}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot 10}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $- 8$ by $10$ .

$x = \frac{5 \pm \sqrt{25 - 80}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Subtract $80$ from $25$ .

$x = \frac{5 \pm \sqrt{- 55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Rewrite $- 55$ as $- 1 (55)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Rewrite $\sqrt{- 1 (55)}$ as $\sqrt{- 1} \cdot \sqrt{55}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

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

$x = \frac{5 \pm i \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

$x = \frac{5 \pm i \sqrt{55}}{2 \cdot 2}$

$y = x^{2} - 3 x + 6$

Multiply $2$ by $2$ .

$x = \frac{5 \pm i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

Change the $\pm$ to $-$ .

$x = \frac{5 - i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{55}}{4}, \frac{5 - i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

$x = \frac{5 + i \sqrt{55}}{4}, \frac{5 - i \sqrt{55}}{4}$

$y = x^{2} - 3 x + 6$

Replace all occurrences of $x$ with $\frac{5 + i \sqrt{55}}{4}$ in each equation.

Replace all occurrences of $x$ in $y = x^{2} - 3 x + 6$ with $\frac{5 + i \sqrt{55}}{4}$ .

$y = {(\frac{5 + i \sqrt{55}}{4})}^{2} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Simplify the right side.

Simplify ${(\frac{5 + i \sqrt{55}}{4})}^{2} - 3 \frac{5 + i \sqrt{55}}{4} + 6$ .

Simplify each term.

Apply the product rule to $\frac{5 + i \sqrt{55}}{4}$ .

$y = \frac{{(5 + i \sqrt{55})}^{2}}{4^{2}} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Raise $4$ to the power of $2$ .

$y = \frac{{(5 + i \sqrt{55})}^{2}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Rewrite ${(5 + i \sqrt{55})}^{2}$ as $(5 + i \sqrt{55}) (5 + i \sqrt{55})$ .

$y = \frac{(5 + i \sqrt{55}) (5 + i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Expand $(5 + i \sqrt{55}) (5 + i \sqrt{55})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{5 (5 + i \sqrt{55}) + i \sqrt{55} (5 + i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Apply the distributive property.

$y = \frac{5 \cdot 5 + 5 (i \sqrt{55}) + i \sqrt{55} (5 + i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Apply the distributive property.

$y = \frac{5 \cdot 5 + 5 (i \sqrt{55}) + i \sqrt{55} \cdot 5 + i \sqrt{55} (i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{5 \cdot 5 + 5 (i \sqrt{55}) + i \sqrt{55} \cdot 5 + i \sqrt{55} (i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$y = \frac{25 + 5 (i \sqrt{55}) + i \sqrt{55} \cdot 5 + i \sqrt{55} (i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Move $5$ to the left of $i \sqrt{55}$ .

$y = \frac{25 + 5 i \sqrt{55} + 5 \cdot (i \sqrt{55}) + i \sqrt{55} (i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Multiply $i \sqrt{55} (i \sqrt{55})$ .

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

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i i \sqrt{55} \sqrt{55}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

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

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i i \sqrt{55} \sqrt{55}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

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

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i^{1 + 1} \sqrt{55} \sqrt{55}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Add $1$ and $1$ .

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i^{2} \sqrt{55} \sqrt{55}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Raise $\sqrt{55}$ to the power of $1$ .

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i^{2} (\sqrt{55} \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Raise $\sqrt{55}$ to the power of $1$ .

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i^{2} (\sqrt{55} \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

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

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i^{2} {\sqrt{55}}^{1 + 1}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Add $1$ and $1$ .

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i^{2} {\sqrt{55}}^{2}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} + i^{2} {\sqrt{55}}^{2}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

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

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 {\sqrt{55}}^{2}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Rewrite ${\sqrt{55}}^{2}$ as $55$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{55}$ as $55^{\frac{1}{2}}$ .

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 {(55^{\frac{1}{2}})}^{2}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

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

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 \cdot 55^{\frac{1}{2} \cdot 2}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

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

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 \cdot 55^{\frac{2}{2}}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 \cdot 55^{\frac{2}{2}}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Rewrite the expression.

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 \cdot 55}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 \cdot 55}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Evaluate the exponent.

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 \cdot 55}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 1 \cdot 55}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Multiply $- 1$ by $55$ .

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 55}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{25 + 5 i \sqrt{55} + 5 i \sqrt{55} - 55}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Subtract $55$ from $25$ .

$y = \frac{5 i \sqrt{55} + 5 i \sqrt{55} - 30}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Add $5 i \sqrt{55}$ and $5 i \sqrt{55}$ .

$y = \frac{10 i \sqrt{55} - 30}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{10 i \sqrt{55} - 30}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Reorder $10 i \sqrt{55}$ and $- 30$ .

$y = \frac{- 30 + 10 i \sqrt{55}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Cancel the common factor of $- 30 + 10 i \sqrt{55}$ and $16$ .

Factor $2$ out of $- 30$ .

$y = \frac{2 (- 15) + 10 i \sqrt{55}}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Factor $2$ out of $10 i \sqrt{55}$ .

$y = \frac{2 (- 15) + 2 (5 i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Factor $2$ out of $2 (- 15) + 2 (5 i \sqrt{55})$ .

$y = \frac{2 (- 15 + 5 i \sqrt{55})}{16} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Cancel the common factors.

Factor $2$ out of $16$ .

$y = \frac{2 (- 15 + 5 i \sqrt{55})}{2 \cdot 8} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Cancel the common factor.

$y = \frac{2 (- 15 + 5 i \sqrt{55})}{2 \cdot 8} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Rewrite the expression.

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - 3 \frac{5 + i \sqrt{55}}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Combine $- 3$ and $\frac{5 + i \sqrt{55}}{4}$ .

$y = \frac{- 15 + 5 i \sqrt{55}}{8} + \frac{- 3 (5 + i \sqrt{55})}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Move the negative in front of the fraction.

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - \frac{3 (5 + i \sqrt{55})}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - \frac{3 (5 + i \sqrt{55})}{4} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

To write $- \frac{3 (5 + i \sqrt{55})}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - \frac{3 (5 + i \sqrt{55})}{4} \cdot \frac{2}{2} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 (5 + i \sqrt{55})}{4}$ by $\frac{2}{2}$ .

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - \frac{3 (5 + i \sqrt{55}) \cdot 2}{4 \cdot 2} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Multiply $4$ by $2$ .

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - \frac{3 (5 + i \sqrt{55}) \cdot 2}{8} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{- 15 + 5 i \sqrt{55}}{8} - \frac{3 (5 + i \sqrt{55}) \cdot 2}{8} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

Combine the numerators over the common denominator.

$y = \frac{- 45 - i \sqrt{55}}{8} + 6$

$x = \frac{5 + i \sqrt{55}}{4}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{- 45 - i \sqrt{55}}{8} + 6 \cdot \frac{8}{8}$

$x = \frac{5 + i \sqrt{55}}{4}$

Combine $6$ and $\frac{8}{8}$ .

$y = \frac{- 45 - i \sqrt{55}}{8} + \frac{6 \cdot 8}{8}$

$x = \frac{5 + i \sqrt{55}}{4}$

Combine the numerators over the common denominator.

$y = \frac{3 - i \sqrt{55}}{8}$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{3 - i \sqrt{55}}{8}$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{3 - i \sqrt{55}}{8}$

$x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{3 - i \sqrt{55}}{8}$

$x = \frac{5 + i \sqrt{55}}{4}$

Replace all occurrences of $x$ with $\frac{5 - i \sqrt{55}}{4}$ in each equation.

Replace all occurrences of $x$ in $y = x^{2} - 3 x + 6$ with $\frac{5 - i \sqrt{55}}{4}$ .

$y = {(\frac{5 - i \sqrt{55}}{4})}^{2} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Simplify the right side.

Simplify ${(\frac{5 - i \sqrt{55}}{4})}^{2} - 3 \frac{5 - i \sqrt{55}}{4} + 6$ .

Simplify each term.

Apply the product rule to $\frac{5 - i \sqrt{55}}{4}$ .

$y = \frac{{(5 - i \sqrt{55})}^{2}}{4^{2}} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Raise $4$ to the power of $2$ .

$y = \frac{{(5 - i \sqrt{55})}^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Rewrite ${(5 - i \sqrt{55})}^{2}$ as $(5 - i \sqrt{55}) (5 - i \sqrt{55})$ .

$y = \frac{(5 - i \sqrt{55}) (5 - i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Expand $(5 - i \sqrt{55}) (5 - i \sqrt{55})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{5 (5 - i \sqrt{55}) - i \sqrt{55} (5 - i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Apply the distributive property.

$y = \frac{5 \cdot 5 + 5 (- i \sqrt{55}) - i \sqrt{55} (5 - i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Apply the distributive property.

$y = \frac{5 \cdot 5 + 5 (- i \sqrt{55}) - i \sqrt{55} \cdot 5 - i \sqrt{55} (- i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{5 \cdot 5 + 5 (- i \sqrt{55}) - i \sqrt{55} \cdot 5 - i \sqrt{55} (- i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$y = \frac{25 + 5 (- i \sqrt{55}) - i \sqrt{55} \cdot 5 - i \sqrt{55} (- i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Multiply $- 1$ by $5$ .

$y = \frac{25 - 5 (i \sqrt{55}) - i \sqrt{55} \cdot 5 - i \sqrt{55} (- i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Multiply $5$ by $- 1$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} - i \sqrt{55} (- i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Multiply $- i \sqrt{55} (- i \sqrt{55})$ .

Multiply $- 1$ by $- 1$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 1 i \sqrt{55} (i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Multiply $\sqrt{55}$ by $1$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + \sqrt{55} i (i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Raise $\sqrt{55}$ to the power of $1$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + \sqrt{55} \sqrt{55} i i}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Raise $\sqrt{55}$ to the power of $1$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + \sqrt{55} \sqrt{55} i i}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

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

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + {\sqrt{55}}^{1 + 1} i i}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Add $1$ and $1$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + {\sqrt{55}}^{2} i i}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

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

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + {\sqrt{55}}^{2} (i i)}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

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

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + {\sqrt{55}}^{2} (i i)}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

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

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + {\sqrt{55}}^{2} i^{1 + 1}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Add $1$ and $1$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + {\sqrt{55}}^{2} i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + {\sqrt{55}}^{2} i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Rewrite ${\sqrt{55}}^{2}$ as $55$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{55}$ as $55^{\frac{1}{2}}$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + {(55^{\frac{1}{2}})}^{2} i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

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

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 55^{\frac{1}{2} \cdot 2} i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

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

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 55^{\frac{2}{2}} i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 55^{\frac{2}{2}} i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Rewrite the expression.

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 55 i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 55 i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Evaluate the exponent.

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 55 i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 55 i^{2}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

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

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} + 55 \cdot - 1}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Multiply $55$ by $- 1$ .

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} - 55}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{25 - 5 i \sqrt{55} - 5 i \sqrt{55} - 55}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Subtract $55$ from $25$ .

$y = \frac{- 5 i \sqrt{55} - 5 i \sqrt{55} - 30}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Subtract $5 i \sqrt{55}$ from $- 5 i \sqrt{55}$ .

$y = \frac{- 10 i \sqrt{55} - 30}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{- 10 i \sqrt{55} - 30}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Reorder $- 10 i \sqrt{55}$ and $- 30$ .

$y = \frac{- 30 - 10 i \sqrt{55}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Cancel the common factor of $- 30 - 10 i \sqrt{55}$ and $16$ .

Factor $2$ out of $- 30$ .

$y = \frac{2 (- 15) - 10 i \sqrt{55}}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Factor $2$ out of $- 10 i \sqrt{55}$ .

$y = \frac{2 (- 15) + 2 (- 5 i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Factor $2$ out of $2 (- 15) + 2 (- 5 i \sqrt{55})$ .

$y = \frac{2 (- 15 - 5 i \sqrt{55})}{16} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Cancel the common factors.

Factor $2$ out of $16$ .

$y = \frac{2 (- 15 - 5 i \sqrt{55})}{2 \cdot 8} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Cancel the common factor.

$y = \frac{2 (- 15 - 5 i \sqrt{55})}{2 \cdot 8} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Rewrite the expression.

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - 3 \frac{5 - i \sqrt{55}}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Combine $- 3$ and $\frac{5 - i \sqrt{55}}{4}$ .

$y = \frac{- 15 - 5 i \sqrt{55}}{8} + \frac{- 3 (5 - i \sqrt{55})}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Move the negative in front of the fraction.

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - \frac{3 (5 - i \sqrt{55})}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - \frac{3 (5 - i \sqrt{55})}{4} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

To write $- \frac{3 (5 - i \sqrt{55})}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - \frac{3 (5 - i \sqrt{55})}{4} \cdot \frac{2}{2} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 (5 - i \sqrt{55})}{4}$ by $\frac{2}{2}$ .

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - \frac{3 (5 - i \sqrt{55}) \cdot 2}{4 \cdot 2} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Multiply $4$ by $2$ .

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - \frac{3 (5 - i \sqrt{55}) \cdot 2}{8} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{- 15 - 5 i \sqrt{55}}{8} - \frac{3 (5 - i \sqrt{55}) \cdot 2}{8} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

Combine the numerators over the common denominator.

$y = \frac{- 45 + i \sqrt{55}}{8} + 6$

$x = \frac{5 - i \sqrt{55}}{4}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{- 45 + i \sqrt{55}}{8} + 6 \cdot \frac{8}{8}$

$x = \frac{5 - i \sqrt{55}}{4}$

Combine $6$ and $\frac{8}{8}$ .

$y = \frac{- 45 + i \sqrt{55}}{8} + \frac{6 \cdot 8}{8}$

$x = \frac{5 - i \sqrt{55}}{4}$

Combine the numerators over the common denominator.

$y = \frac{3 + i \sqrt{55}}{8}$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{3 + i \sqrt{55}}{8}$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{3 + i \sqrt{55}}{8}$

$x = \frac{5 - i \sqrt{55}}{4}$

$y = \frac{3 + i \sqrt{55}}{8}$

$x = \frac{5 - i \sqrt{55}}{4}$

List all of the solutions.

$y = \frac{3 - i \sqrt{55}}{8}, x = \frac{5 + i \sqrt{55}}{4}$

$y = \frac{3 + i \sqrt{55}}{8}, x = \frac{5 - i \sqrt{55}}{4}$

Solve the System of Equations y-6=x^2-3x x+2y=2

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