Solve the System of Equations y=x-8 y=x^2-2x-3

$y = x - 8$ $y = x^{2} - 2 x - 3$

Step 1

Eliminate the equal sides of each equation and combine.

$x - 8 = x^{2} - 2 x - 3$

Step 2

Solve $x - 8 = x^{2} - 2 x - 3$ 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.

$x^{2} - 2 x - 3 = x - 8$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 2 x - 3 - x = - 8$

Subtract $x$ from $- 2 x$ .

$x^{2} - 3 x - 3 = - 8$

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

Add $8$ to both sides of the equation.

$x^{2} - 3 x - 3 + 8 = 0$

Add $- 3$ and $8$ .

$x^{2} - 3 x + 5 = 0$

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 $a = 1$ , $b = - 3$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1} + y = x^{2} - 2 x - 3$

Simplify.

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{3 \pm \sqrt{9 - 20}}{2 \cdot 1}$

Subtract $20$ from $9$ .

$x = \frac{3 \pm \sqrt{- 11}}{2 \cdot 1}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

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

$x = \frac{3 \pm i \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm i \sqrt{11}}{2}$

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

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{3 \pm \sqrt{9 - 20}}{2 \cdot 1}$

Subtract $20$ from $9$ .

$x = \frac{3 \pm \sqrt{- 11}}{2 \cdot 1}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

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

$x = \frac{3 \pm i \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm i \sqrt{11}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{3 + i \sqrt{11}}{2}$

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

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{3 \pm \sqrt{9 - 20}}{2 \cdot 1}$

Subtract $20$ from $9$ .

$x = \frac{3 \pm \sqrt{- 11}}{2 \cdot 1}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

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

$x = \frac{3 \pm i \sqrt{11}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm i \sqrt{11}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{3 - i \sqrt{11}}{2}$

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{11}}{2}, \frac{3 - i \sqrt{11}}{2}$

Step 3

Evaluate $y$ when $x = \frac{3 + i \sqrt{11}}{2}$ .

Substitute $\frac{3 + i \sqrt{11}}{2}$ for $x$ .

$y = {(\frac{3 + i \sqrt{11}}{2})}^{2} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Simplify ${(\frac{3 + i \sqrt{11}}{2})}^{2} - 2 \frac{3 + i \sqrt{11}}{2} - 3$ .

Simplify each term.

Apply the product rule to $\frac{3 + i \sqrt{11}}{2}$ .

$y = \frac{{(3 + i \sqrt{11})}^{2}}{2^{2}} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

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

$y = \frac{{(3 + i \sqrt{11})}^{2}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Rewrite ${(3 + i \sqrt{11})}^{2}$ as $(3 + i \sqrt{11}) (3 + i \sqrt{11})$ .

$y = \frac{(3 + i \sqrt{11}) (3 + i \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Expand $(3 + i \sqrt{11}) (3 + i \sqrt{11})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{3 (3 + i \sqrt{11}) + i \sqrt{11} (3 + i \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Apply the distributive property.

$y = \frac{3 \cdot 3 + 3 (i \sqrt{11}) + i \sqrt{11} (3 + i \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Apply the distributive property.

$y = \frac{3 \cdot 3 + 3 (i \sqrt{11}) + i \sqrt{11} \cdot 3 + i \sqrt{11} (i \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$y = \frac{9 + 3 (i \sqrt{11}) + i \sqrt{11} \cdot 3 + i \sqrt{11} (i \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Move $3$ to the left of $i \sqrt{11}$ .

$y = \frac{9 + 3 i \sqrt{11} + 3 \cdot (i \sqrt{11}) + i \sqrt{11} (i \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Multiply $i \sqrt{11} (i \sqrt{11})$ .

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

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} + i^{1} i \sqrt{11} \sqrt{11}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} + i^{1} i^{1} \sqrt{11} \sqrt{11}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} + i^{1 + 1} \sqrt{11} \sqrt{11}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} + i^{2} \sqrt{11} \sqrt{11}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Raise $\sqrt{11}$ to the power of $1$ .

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} + i^{2} ({\sqrt{11}}^{1} \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Raise $\sqrt{11}$ to the power of $1$ .

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} + i^{2} ({\sqrt{11}}^{1} {\sqrt{11}}^{1})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} + i^{2} {\sqrt{11}}^{1 + 1}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} + i^{2} {\sqrt{11}}^{2}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} - 1 {\sqrt{11}}^{2}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} - 1 {(11^{\frac{1}{2}})}^{2}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} - 1 \cdot 11^{\frac{1}{2} \cdot 2}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} - 1 \cdot 11^{\frac{2}{2}}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} - 1 \cdot 11^{\frac{2}{2}}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Rewrite the expression.

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} - 1 \cdot 11^{1}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Evaluate the exponent.

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} - 1 \cdot 11}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Multiply $- 1$ by $11$ .

$y = \frac{9 + 3 i \sqrt{11} + 3 i \sqrt{11} - 11}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Subtract $11$ from $9$ .

$y = \frac{3 i \sqrt{11} + 3 i \sqrt{11} - 2}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Add $3 i \sqrt{11}$ and $3 i \sqrt{11}$ .

$y = \frac{6 i \sqrt{11} - 2}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Reorder $6 i \sqrt{11}$ and $- 2$ .

$y = \frac{- 2 + 6 i \sqrt{11}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Cancel the common factor of $- 2 + 6 i \sqrt{11}$ and $4$ .

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1) + 6 i \sqrt{11}}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Factor $2$ out of $6 i \sqrt{11}$ .

$y = \frac{2 (- 1) + 2 (3 i \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Factor $2$ out of $2 (- 1) + 2 (3 i \sqrt{11})$ .

$y = \frac{2 (- 1 + 3 i \sqrt{11})}{4} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (- 1 + 3 i \sqrt{11})}{2 \cdot 2} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Cancel the common factor.

$y = \frac{2 (- 1 + 3 i \sqrt{11})}{2 \cdot 2} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Rewrite the expression.

$y = \frac{- 1 + 3 i \sqrt{11}}{2} - 2 \frac{3 + i \sqrt{11}}{2} - 3$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$y = \frac{- 1 + 3 i \sqrt{11}}{2} + 2 (- 1) \frac{3 + i \sqrt{11}}{2} - 3$

Cancel the common factor.

$y = \frac{- 1 + 3 i \sqrt{11}}{2} + 2 \cdot - 1 \frac{3 + i \sqrt{11}}{2} - 3$

Rewrite the expression.

$y = \frac{- 1 + 3 i \sqrt{11}}{2} - 1 (3 + i \sqrt{11}) - 3$

Apply the distributive property.

$y = \frac{- 1 + 3 i \sqrt{11}}{2} - 1 \cdot 3 - 1 (i \sqrt{11}) - 3$

Multiply $- 1$ by $3$ .

$y = \frac{- 1 + 3 i \sqrt{11}}{2} - 3 - 1 (i \sqrt{11}) - 3$

Rewrite $- 1 (i \sqrt{11})$ as $- (i \sqrt{11})$ .

$y = \frac{- 1 + 3 i \sqrt{11}}{2} - 3 - i \sqrt{11} - 3$

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

$y = \frac{- 1 + 3 i \sqrt{11}}{2} - 3 \cdot \frac{2}{2} - i \sqrt{11} - 3$

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

$y = \frac{- 1 + 3 i \sqrt{11}}{2} + \frac{- 3 \cdot 2}{2} - i \sqrt{11} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 7 + 3 i \sqrt{11}}{2} - i \sqrt{11} - 3$

To write $- i \sqrt{11}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 7 + 3 i \sqrt{11}}{2} - i \sqrt{11} \cdot \frac{2}{2} - 3$

Combine $- i \sqrt{11}$ and $\frac{2}{2}$ .

$y = \frac{- 7 + 3 i \sqrt{11}}{2} + \frac{- i \sqrt{11} \cdot 2}{2} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 7 + i \sqrt{11}}{2} - 3$

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

$y = \frac{- 7 + i \sqrt{11}}{2} - 3 \cdot \frac{2}{2}$

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

$y = \frac{- 7 + i \sqrt{11}}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 13 + i \sqrt{11}}{2}$

Rewrite $- 13$ as $- 1 (13)$ .

$y = \frac{- 1 (13) + i \sqrt{11}}{2}$

Factor $- 1$ out of $i \sqrt{11}$ .

$y = \frac{- 1 (13) - (- i \sqrt{11})}{2}$

Factor $- 1$ out of $- 1 (13) - (- i \sqrt{11})$ .

$y = \frac{- 1 (13 - i \sqrt{11})}{2}$

Move the negative in front of the fraction.

$y = - \frac{13 - i \sqrt{11}}{2}$

Step 4

Evaluate $y$ when $x = \frac{3 - i \sqrt{11}}{2}$ .

Substitute $\frac{3 - i \sqrt{11}}{2}$ for $x$ .

$y = {(\frac{3 - i \sqrt{11}}{2})}^{2} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Simplify ${(\frac{3 - i \sqrt{11}}{2})}^{2} - 2 \frac{3 - i \sqrt{11}}{2} - 3$ .

Simplify each term.

Apply the product rule to $\frac{3 - i \sqrt{11}}{2}$ .

$y = \frac{{(3 - i \sqrt{11})}^{2}}{2^{2}} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

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

$y = \frac{{(3 - i \sqrt{11})}^{2}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Rewrite ${(3 - i \sqrt{11})}^{2}$ as $(3 - i \sqrt{11}) (3 - i \sqrt{11})$ .

$y = \frac{(3 - i \sqrt{11}) (3 - i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Expand $(3 - i \sqrt{11}) (3 - i \sqrt{11})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{3 (3 - i \sqrt{11}) - i \sqrt{11} (3 - i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Apply the distributive property.

$y = \frac{3 \cdot 3 + 3 (- i \sqrt{11}) - i \sqrt{11} (3 - i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Apply the distributive property.

$y = \frac{3 \cdot 3 + 3 (- i \sqrt{11}) - i \sqrt{11} \cdot 3 - i \sqrt{11} (- i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$y = \frac{9 + 3 (- i \sqrt{11}) - i \sqrt{11} \cdot 3 - i \sqrt{11} (- i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Multiply $- 1$ by $3$ .

$y = \frac{9 - 3 (i \sqrt{11}) - i \sqrt{11} \cdot 3 - i \sqrt{11} (- i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Multiply $3$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} - i \sqrt{11} (- i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Multiply $- i \sqrt{11} (- i \sqrt{11})$ .

Multiply $- 1$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + 1 i \sqrt{11} (i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Multiply $\sqrt{11}$ by $1$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + \sqrt{11} i (i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Raise $\sqrt{11}$ to the power of $1$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {\sqrt{11}}^{1} \sqrt{11} i i}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Raise $\sqrt{11}$ to the power of $1$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {\sqrt{11}}^{1} {\sqrt{11}}^{1} i i}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {\sqrt{11}}^{1 + 1} i i}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {\sqrt{11}}^{2} i i}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {\sqrt{11}}^{2} (i^{1} i)}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {\sqrt{11}}^{2} (i^{1} i^{1})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {\sqrt{11}}^{2} i^{1 + 1}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {\sqrt{11}}^{2} i^{2}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + {(11^{\frac{1}{2}})}^{2} i^{2}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + 11^{\frac{1}{2} \cdot 2} i^{2}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + 11^{\frac{2}{2}} i^{2}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + 11^{\frac{2}{2}} i^{2}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Rewrite the expression.

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + 11^{1} i^{2}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Evaluate the exponent.

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + 11 i^{2}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} + 11 \cdot - 1}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Multiply $11$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{11} - 3 i \sqrt{11} - 11}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Subtract $11$ from $9$ .

$y = \frac{- 3 i \sqrt{11} - 3 i \sqrt{11} - 2}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Subtract $3 i \sqrt{11}$ from $- 3 i \sqrt{11}$ .

$y = \frac{- 6 i \sqrt{11} - 2}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Reorder $- 6 i \sqrt{11}$ and $- 2$ .

$y = \frac{- 2 - 6 i \sqrt{11}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Cancel the common factor of $- 2 - 6 i \sqrt{11}$ and $4$ .

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1) - 6 i \sqrt{11}}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Factor $2$ out of $- 6 i \sqrt{11}$ .

$y = \frac{2 (- 1) + 2 (- 3 i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Factor $2$ out of $2 (- 1) + 2 (- 3 i \sqrt{11})$ .

$y = \frac{2 (- 1 - 3 i \sqrt{11})}{4} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (- 1 - 3 i \sqrt{11})}{2 \cdot 2} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Cancel the common factor.

$y = \frac{2 (- 1 - 3 i \sqrt{11})}{2 \cdot 2} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Rewrite the expression.

$y = \frac{- 1 - 3 i \sqrt{11}}{2} - 2 \frac{3 - i \sqrt{11}}{2} - 3$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$y = \frac{- 1 - 3 i \sqrt{11}}{2} + 2 (- 1) \frac{3 - i \sqrt{11}}{2} - 3$

Cancel the common factor.

$y = \frac{- 1 - 3 i \sqrt{11}}{2} + 2 \cdot - 1 \frac{3 - i \sqrt{11}}{2} - 3$

Rewrite the expression.

$y = \frac{- 1 - 3 i \sqrt{11}}{2} - 1 (3 - i \sqrt{11}) - 3$

Apply the distributive property.

$y = \frac{- 1 - 3 i \sqrt{11}}{2} - 1 \cdot 3 - 1 (- i \sqrt{11}) - 3$

Multiply $- 1$ by $3$ .

$y = \frac{- 1 - 3 i \sqrt{11}}{2} - 3 - 1 (- i \sqrt{11}) - 3$

Multiply $- 1 (- i \sqrt{11})$ .

Multiply $- 1$ by $- 1$ .

$y = \frac{- 1 - 3 i \sqrt{11}}{2} - 3 + 1 (i \sqrt{11}) - 3$

Multiply $\sqrt{11}$ by $1$ .

$y = \frac{- 1 - 3 i \sqrt{11}}{2} - 3 + \sqrt{11} i - 3$

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

$y = \frac{- 1 - 3 i \sqrt{11}}{2} - 3 \cdot \frac{2}{2} + \sqrt{11} i - 3$

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

$y = \frac{- 1 - 3 i \sqrt{11}}{2} + \frac{- 3 \cdot 2}{2} + \sqrt{11} i - 3$

Combine the numerators over the common denominator.

$y = \frac{- 7 - 3 i \sqrt{11}}{2} + \sqrt{11} i - 3$

To write $\sqrt{11} i$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 7 - 3 i \sqrt{11}}{2} + \sqrt{11} i \cdot \frac{2}{2} - 3$

Combine $\sqrt{11} i$ and $\frac{2}{2}$ .

$y = \frac{- 7 - 3 i \sqrt{11}}{2} + \frac{\sqrt{11} i \cdot 2}{2} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 7 - i \sqrt{11}}{2} - 3$

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

$y = \frac{- 7 - i \sqrt{11}}{2} - 3 \cdot \frac{2}{2}$

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

$y = \frac{- 7 - i \sqrt{11}}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 13 - i \sqrt{11}}{2}$

Rewrite $- 13$ as $- 1 (13)$ .

$y = \frac{- 1 (13) - i \sqrt{11}}{2}$

Factor $- 1$ out of $- i \sqrt{11}$ .

$y = \frac{- 1 (13) - (i \sqrt{11})}{2}$

Factor $- 1$ out of $- 1 (13) - (i \sqrt{11})$ .

$y = \frac{- 1 (13 + i \sqrt{11})}{2}$

Move the negative in front of the fraction.

$y = - \frac{13 + i \sqrt{11}}{2}$

Step 5

List all of the solutions.

$y = - \frac{13 - i \sqrt{11}}{2}, x = \frac{3 + i \sqrt{11}}{2}$

$y = - \frac{13 + i \sqrt{11}}{2}, x = \frac{3 - i \sqrt{11}}{2}$

Step 6

Solve the System of Equations y=x-8 y=x^2-2x-3

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