Solve the System of Equations y=-x-10 y=x^2-2x-3

$y = - x - 10$ $y = x^{2} - 2 x - 3$

Eliminate the equal sides of each equation and combine.

$- x - 10 = x^{2} - 2 x - 3$

Solve $- x - 10 = x^{2} - 2 x - 3$ for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 2 x - 3 = - x - 10$

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$x^{2} - 2 x - 3 + x = - 10$

Add $- 2 x$ and $x$ .

$x^{2} - x - 3 = - 10$

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

Add $10$ to both sides of the equation.

$x^{2} - x - 3 + 10 = 0$

Add $- 3$ and $10$ .

$x^{2} - x + 7 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = x^{2} - 2 x - 3$

Substitute the values $a = 1$ , $b = - 1$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 7)}}{2 \cdot 1} + y = x^{2} - 2 x - 3$

Simplify.

Simplify the numerator.

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $7$ .

$x = \frac{1 \pm \sqrt{1 - 28}}{2 \cdot 1}$

Subtract $28$ from $1$ .

$x = \frac{1 \pm \sqrt{- 27}}{2 \cdot 1}$

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$x = \frac{1 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

Move $3$ to the left of $i$ .

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

Multiply $2$ by $1$ .

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

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

Simplify the numerator.

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $7$ .

$x = \frac{1 \pm \sqrt{1 - 28}}{2 \cdot 1}$

Subtract $28$ from $1$ .

$x = \frac{1 \pm \sqrt{- 27}}{2 \cdot 1}$

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$x = \frac{1 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

Move $3$ to the left of $i$ .

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

Multiply $2$ by $1$ .

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

Change the $\pm$ to $+$ .

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

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

Simplify the numerator.

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $7$ .

$x = \frac{1 \pm \sqrt{1 - 28}}{2 \cdot 1}$

Subtract $28$ from $1$ .

$x = \frac{1 \pm \sqrt{- 27}}{2 \cdot 1}$

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

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

$x = \frac{1 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$x = \frac{1 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

Move $3$ to the left of $i$ .

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

Multiply $2$ by $1$ .

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

Evaluate $y$ when $x = \frac{1 + 3 i \sqrt{3}}{2}$ .

Substitute $\frac{1 + 3 i \sqrt{3}}{2}$ for $x$ .

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

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

Simplify each term.

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

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

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

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

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

Multiply $3$ by $1$ .

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

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

Multiply $3$ by $3$ .

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 i \sqrt{3} (i \sqrt{3})}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 (i^{1} i) \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 i^{1 + 1} \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 i^{2} \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{1 + 1}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} + 9 \cdot - 1 {\sqrt{3}}^{2}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Multiply $9$ by $- 1$ .

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} - 9 {\sqrt{3}}^{2}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} - 9 {(3^{\frac{1}{2}})}^{2}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} - 9 \cdot 3^{\frac{1}{2} \cdot 2}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Rewrite the expression.

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} - 9 \cdot 3^{1}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Evaluate the exponent.

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} - 9 \cdot 3}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Multiply $- 9$ by $3$ .

$y = \frac{1 + 3 i \sqrt{3} + 3 i \sqrt{3} - 27}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Subtract $27$ from $1$ .

$y = \frac{3 i \sqrt{3} + 3 i \sqrt{3} - 26}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{6 i \sqrt{3} - 26}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Reorder $6 i \sqrt{3}$ and $- 26$ .

$y = \frac{- 26 + 6 i \sqrt{3}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Cancel the common factor of $- 26 + 6 i \sqrt{3}$ and $4$ .

Factor $2$ out of $- 26$ .

$y = \frac{2 (- 13) + 6 i \sqrt{3}}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Factor $2$ out of $6 i \sqrt{3}$ .

$y = \frac{2 (- 13) + 2 (3 i \sqrt{3})}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Factor $2$ out of $2 (- 13) + 2 (3 i \sqrt{3})$ .

$y = \frac{2 (- 13 + 3 i \sqrt{3})}{4} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (- 13 + 3 i \sqrt{3})}{2 \cdot 2} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Cancel the common factor.

$y = \frac{2 (- 13 + 3 i \sqrt{3})}{2 \cdot 2} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Rewrite the expression.

$y = \frac{- 13 + 3 i \sqrt{3}}{2} - 2 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$y = \frac{- 13 + 3 i \sqrt{3}}{2} + 2 (- 1) \frac{1 + 3 i \sqrt{3}}{2} - 3$

Cancel the common factor.

$y = \frac{- 13 + 3 i \sqrt{3}}{2} + 2 \cdot - 1 \frac{1 + 3 i \sqrt{3}}{2} - 3$

Rewrite the expression.

$y = \frac{- 13 + 3 i \sqrt{3}}{2} - 1 (1 + 3 i \sqrt{3}) - 3$

Apply the distributive property.

$y = \frac{- 13 + 3 i \sqrt{3}}{2} - 1 \cdot 1 - 1 (3 i \sqrt{3}) - 3$

Multiply $- 1$ by $1$ .

$y = \frac{- 13 + 3 i \sqrt{3}}{2} - 1 - 1 (3 i \sqrt{3}) - 3$

Multiply $3$ by $- 1$ .

$y = \frac{- 13 + 3 i \sqrt{3}}{2} - 1 - 3 i \sqrt{3} - 3$

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

$y = \frac{- 13 + 3 i \sqrt{3}}{2} - 1 \cdot \frac{2}{2} - 3 i \sqrt{3} - 3$

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

$y = \frac{- 13 + 3 i \sqrt{3}}{2} + \frac{- 1 \cdot 2}{2} - 3 i \sqrt{3} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 15 + 3 i \sqrt{3}}{2} - 3 i \sqrt{3} - 3$

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

$y = \frac{- 15 + 3 i \sqrt{3}}{2} - 3 i \sqrt{3} \cdot \frac{2}{2} - 3$

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

$y = \frac{- 15 + 3 i \sqrt{3}}{2} + \frac{- 3 i \sqrt{3} \cdot 2}{2} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 15 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{- 15 - 3 i \sqrt{3}}{2} - 3 \cdot \frac{2}{2}$

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

$y = \frac{- 15 - 3 i \sqrt{3}}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 21 - 3 i \sqrt{3}}{2}$

Rewrite $- 21$ as $- 1 (21)$ .

$y = \frac{- 1 (21) - 3 i \sqrt{3}}{2}$

Factor $- 1$ out of $- 3 i \sqrt{3}$ .

$y = \frac{- 1 (21) - (3 i \sqrt{3})}{2}$

Factor $- 1$ out of $- 1 (21) - (3 i \sqrt{3})$ .

$y = \frac{- 1 (21 + 3 i \sqrt{3})}{2}$

Move the negative in front of the fraction.

$y = - \frac{21 + 3 i \sqrt{3}}{2}$

Evaluate $y$ when $x = \frac{1 - 3 i \sqrt{3}}{2}$ .

Substitute $\frac{1 - 3 i \sqrt{3}}{2}$ for $x$ .

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

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

Simplify each term.

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

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

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

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

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

Multiply $- 3$ by $1$ .

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

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

Multiply $- 3$ by $- 3$ .

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 i \sqrt{3} (i \sqrt{3})}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 (i^{1} i) \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 i^{1 + 1} \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 i^{2} \sqrt{3} \sqrt{3}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{1 + 1}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 i^{2} {\sqrt{3}}^{2}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} + 9 \cdot - 1 {\sqrt{3}}^{2}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Multiply $9$ by $- 1$ .

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} - 9 {\sqrt{3}}^{2}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} - 9 {(3^{\frac{1}{2}})}^{2}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} - 9 \cdot 3^{\frac{1}{2} \cdot 2}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} - 9 \cdot 3^{\frac{2}{2}}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Rewrite the expression.

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} - 9 \cdot 3^{1}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Evaluate the exponent.

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} - 9 \cdot 3}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Multiply $- 9$ by $3$ .

$y = \frac{1 - 3 i \sqrt{3} - 3 i \sqrt{3} - 27}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Subtract $27$ from $1$ .

$y = \frac{- 3 i \sqrt{3} - 3 i \sqrt{3} - 26}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{- 6 i \sqrt{3} - 26}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Reorder $- 6 i \sqrt{3}$ and $- 26$ .

$y = \frac{- 26 - 6 i \sqrt{3}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Cancel the common factor of $- 26 - 6 i \sqrt{3}$ and $4$ .

Factor $2$ out of $- 26$ .

$y = \frac{2 (- 13) - 6 i \sqrt{3}}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Factor $2$ out of $- 6 i \sqrt{3}$ .

$y = \frac{2 (- 13) + 2 (- 3 i \sqrt{3})}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Factor $2$ out of $2 (- 13) + 2 (- 3 i \sqrt{3})$ .

$y = \frac{2 (- 13 - 3 i \sqrt{3})}{4} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (- 13 - 3 i \sqrt{3})}{2 \cdot 2} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Cancel the common factor.

$y = \frac{2 (- 13 - 3 i \sqrt{3})}{2 \cdot 2} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Rewrite the expression.

$y = \frac{- 13 - 3 i \sqrt{3}}{2} - 2 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$y = \frac{- 13 - 3 i \sqrt{3}}{2} + 2 (- 1) \frac{1 - 3 i \sqrt{3}}{2} - 3$

Cancel the common factor.

$y = \frac{- 13 - 3 i \sqrt{3}}{2} + 2 \cdot - 1 \frac{1 - 3 i \sqrt{3}}{2} - 3$

Rewrite the expression.

$y = \frac{- 13 - 3 i \sqrt{3}}{2} - 1 (1 - 3 i \sqrt{3}) - 3$

Apply the distributive property.

$y = \frac{- 13 - 3 i \sqrt{3}}{2} - 1 \cdot 1 - 1 (- 3 i \sqrt{3}) - 3$

Multiply $- 1$ by $1$ .

$y = \frac{- 13 - 3 i \sqrt{3}}{2} - 1 - 1 (- 3 i \sqrt{3}) - 3$

Multiply $- 3$ by $- 1$ .

$y = \frac{- 13 - 3 i \sqrt{3}}{2} - 1 + 3 i \sqrt{3} - 3$

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

$y = \frac{- 13 - 3 i \sqrt{3}}{2} - 1 \cdot \frac{2}{2} + 3 i \sqrt{3} - 3$

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

$y = \frac{- 13 - 3 i \sqrt{3}}{2} + \frac{- 1 \cdot 2}{2} + 3 i \sqrt{3} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 15 - 3 i \sqrt{3}}{2} + 3 i \sqrt{3} - 3$

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

$y = \frac{- 15 - 3 i \sqrt{3}}{2} + 3 i \sqrt{3} \cdot \frac{2}{2} - 3$

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

$y = \frac{- 15 - 3 i \sqrt{3}}{2} + \frac{3 i \sqrt{3} \cdot 2}{2} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 15 + 3 i \sqrt{3}}{2} - 3$

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

$y = \frac{- 15 + 3 i \sqrt{3}}{2} - 3 \cdot \frac{2}{2}$

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

$y = \frac{- 15 + 3 i \sqrt{3}}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 21 + 3 i \sqrt{3}}{2}$

Rewrite $- 21$ as $- 1 (21)$ .

$y = \frac{- 1 (21) + 3 i \sqrt{3}}{2}$

Factor $- 1$ out of $3 i \sqrt{3}$ .

$y = \frac{- 1 (21) - (- 3 i \sqrt{3})}{2}$

Factor $- 1$ out of $- 1 (21) - (- 3 i \sqrt{3})$ .

$y = \frac{- 1 (21 - 3 i \sqrt{3})}{2}$

Move the negative in front of the fraction.

$y = - \frac{21 - 3 i \sqrt{3}}{2}$

List all of the solutions.

$y = - \frac{21 + 3 i \sqrt{3}}{2}, x = \frac{1 + 3 i \sqrt{3}}{2}$

$y = - \frac{21 - 3 i \sqrt{3}}{2}, x = \frac{1 - 3 i \sqrt{3}}{2}$

Solve the System of Equations y=-x-10 y=x^2-2x-3

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