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

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

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.

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $- 2 x$ .

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

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

Add $10$ to both sides of the equation.

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

Add $- 3$ and $10$ .

$x^{2} - 3 x + 7 = 0$

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 $a = 1$ , $b = - 3$ , and $c = 7$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 7$ .

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $7$ .

$x = \frac{3 \pm \sqrt{9 - 28}}{2 \cdot 1}$

Subtract $28$ from $9$ .

$x = \frac{3 \pm \sqrt{- 19}}{2 \cdot 1}$

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 19}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (19)}$ as $\sqrt{- 1} \cdot \sqrt{19}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{19}}{2 \cdot 1}$

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

$x = \frac{3 \pm i \sqrt{19}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm i \sqrt{19}}{2}$

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

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 7$ .

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $7$ .

$x = \frac{3 \pm \sqrt{9 - 28}}{2 \cdot 1}$

Subtract $28$ from $9$ .

$x = \frac{3 \pm \sqrt{- 19}}{2 \cdot 1}$

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 19}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (19)}$ as $\sqrt{- 1} \cdot \sqrt{19}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{19}}{2 \cdot 1}$

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

$x = \frac{3 \pm i \sqrt{19}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm i \sqrt{19}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{3 + i \sqrt{19}}{2}$

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

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 7$ .

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $7$ .

$x = \frac{3 \pm \sqrt{9 - 28}}{2 \cdot 1}$

Subtract $28$ from $9$ .

$x = \frac{3 \pm \sqrt{- 19}}{2 \cdot 1}$

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 19}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (19)}$ as $\sqrt{- 1} \cdot \sqrt{19}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{19}}{2 \cdot 1}$

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

$x = \frac{3 \pm i \sqrt{19}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm i \sqrt{19}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{3 - i \sqrt{19}}{2}$

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{19}}{2}, \frac{3 - i \sqrt{19}}{2}$

Step 3

Evaluate $y$ when $x = \frac{3 + i \sqrt{19}}{2}$ .

Substitute $\frac{3 + i \sqrt{19}}{2}$ for $x$ .

$y = {(\frac{3 + i \sqrt{19}}{2})}^{2} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Simplify ${(\frac{3 + i \sqrt{19}}{2})}^{2} - 2 \frac{3 + i \sqrt{19}}{2} - 3$ .

Simplify each term.

Apply the product rule to $\frac{3 + i \sqrt{19}}{2}$ .

$y = \frac{{(3 + i \sqrt{19})}^{2}}{2^{2}} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

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

$y = \frac{{(3 + i \sqrt{19})}^{2}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Rewrite ${(3 + i \sqrt{19})}^{2}$ as $(3 + i \sqrt{19}) (3 + i \sqrt{19})$ .

$y = \frac{(3 + i \sqrt{19}) (3 + i \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Expand $(3 + i \sqrt{19}) (3 + i \sqrt{19})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{3 (3 + i \sqrt{19}) + i \sqrt{19} (3 + i \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Apply the distributive property.

$y = \frac{3 \cdot 3 + 3 (i \sqrt{19}) + i \sqrt{19} (3 + i \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Apply the distributive property.

$y = \frac{3 \cdot 3 + 3 (i \sqrt{19}) + i \sqrt{19} \cdot 3 + i \sqrt{19} (i \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$y = \frac{9 + 3 (i \sqrt{19}) + i \sqrt{19} \cdot 3 + i \sqrt{19} (i \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Move $3$ to the left of $i \sqrt{19}$ .

$y = \frac{9 + 3 i \sqrt{19} + 3 \cdot (i \sqrt{19}) + i \sqrt{19} (i \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Multiply $i \sqrt{19} (i \sqrt{19})$ .

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

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{1} i \sqrt{19} \sqrt{19}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{1} i^{1} \sqrt{19} \sqrt{19}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{1 + 1} \sqrt{19} \sqrt{19}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} \sqrt{19} \sqrt{19}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Raise $\sqrt{19}$ to the power of $1$ .

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} ({\sqrt{19}}^{1} \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Raise $\sqrt{19}$ to the power of $1$ .

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} ({\sqrt{19}}^{1} {\sqrt{19}}^{1})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} {\sqrt{19}}^{1 + 1}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} {\sqrt{19}}^{2}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 {\sqrt{19}}^{2}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 {(19^{\frac{1}{2}})}^{2}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19^{\frac{1}{2} \cdot 2}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

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

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19^{\frac{2}{2}}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19^{\frac{2}{2}}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Rewrite the expression.

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19^{1}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Evaluate the exponent.

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Multiply $- 1$ by $19$ .

$y = \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 19}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Subtract $19$ from $9$ .

$y = \frac{3 i \sqrt{19} + 3 i \sqrt{19} - 10}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Add $3 i \sqrt{19}$ and $3 i \sqrt{19}$ .

$y = \frac{6 i \sqrt{19} - 10}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Reorder $6 i \sqrt{19}$ and $- 10$ .

$y = \frac{- 10 + 6 i \sqrt{19}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Cancel the common factor of $- 10 + 6 i \sqrt{19}$ and $4$ .

Factor $2$ out of $- 10$ .

$y = \frac{2 (- 5) + 6 i \sqrt{19}}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Factor $2$ out of $6 i \sqrt{19}$ .

$y = \frac{2 (- 5) + 2 (3 i \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Factor $2$ out of $2 (- 5) + 2 (3 i \sqrt{19})$ .

$y = \frac{2 (- 5 + 3 i \sqrt{19})}{4} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (- 5 + 3 i \sqrt{19})}{2 \cdot 2} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Cancel the common factor.

$y = \frac{2 (- 5 + 3 i \sqrt{19})}{2 \cdot 2} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Rewrite the expression.

$y = \frac{- 5 + 3 i \sqrt{19}}{2} - 2 \frac{3 + i \sqrt{19}}{2} - 3$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$y = \frac{- 5 + 3 i \sqrt{19}}{2} + 2 (- 1) \frac{3 + i \sqrt{19}}{2} - 3$

Cancel the common factor.

$y = \frac{- 5 + 3 i \sqrt{19}}{2} + 2 \cdot - 1 \frac{3 + i \sqrt{19}}{2} - 3$

Rewrite the expression.

$y = \frac{- 5 + 3 i \sqrt{19}}{2} - 1 (3 + i \sqrt{19}) - 3$

Apply the distributive property.

$y = \frac{- 5 + 3 i \sqrt{19}}{2} - 1 \cdot 3 - 1 (i \sqrt{19}) - 3$

Multiply $- 1$ by $3$ .

$y = \frac{- 5 + 3 i \sqrt{19}}{2} - 3 - 1 (i \sqrt{19}) - 3$

Rewrite $- 1 (i \sqrt{19})$ as $- (i \sqrt{19})$ .

$y = \frac{- 5 + 3 i \sqrt{19}}{2} - 3 - i \sqrt{19} - 3$

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

$y = \frac{- 5 + 3 i \sqrt{19}}{2} - 3 \cdot \frac{2}{2} - i \sqrt{19} - 3$

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

$y = \frac{- 5 + 3 i \sqrt{19}}{2} + \frac{- 3 \cdot 2}{2} - i \sqrt{19} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 11 + 3 i \sqrt{19}}{2} - i \sqrt{19} - 3$

To write $- i \sqrt{19}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 11 + 3 i \sqrt{19}}{2} - i \sqrt{19} \cdot \frac{2}{2} - 3$

Combine $- i \sqrt{19}$ and $\frac{2}{2}$ .

$y = \frac{- 11 + 3 i \sqrt{19}}{2} + \frac{- i \sqrt{19} \cdot 2}{2} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 11 + i \sqrt{19}}{2} - 3$

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

$y = \frac{- 11 + i \sqrt{19}}{2} - 3 \cdot \frac{2}{2}$

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

$y = \frac{- 11 + i \sqrt{19}}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 17 + i \sqrt{19}}{2}$

Rewrite $- 17$ as $- 1 (17)$ .

$y = \frac{- 1 (17) + i \sqrt{19}}{2}$

Factor $- 1$ out of $i \sqrt{19}$ .

$y = \frac{- 1 (17) - (- i \sqrt{19})}{2}$

Factor $- 1$ out of $- 1 (17) - (- i \sqrt{19})$ .

$y = \frac{- 1 (17 - i \sqrt{19})}{2}$

Move the negative in front of the fraction.

$y = - \frac{17 - i \sqrt{19}}{2}$

Step 4

Evaluate $y$ when $x = \frac{3 - i \sqrt{19}}{2}$ .

Substitute $\frac{3 - i \sqrt{19}}{2}$ for $x$ .

$y = {(\frac{3 - i \sqrt{19}}{2})}^{2} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Simplify ${(\frac{3 - i \sqrt{19}}{2})}^{2} - 2 \frac{3 - i \sqrt{19}}{2} - 3$ .

Simplify each term.

Apply the product rule to $\frac{3 - i \sqrt{19}}{2}$ .

$y = \frac{{(3 - i \sqrt{19})}^{2}}{2^{2}} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

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

$y = \frac{{(3 - i \sqrt{19})}^{2}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Rewrite ${(3 - i \sqrt{19})}^{2}$ as $(3 - i \sqrt{19}) (3 - i \sqrt{19})$ .

$y = \frac{(3 - i \sqrt{19}) (3 - i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Expand $(3 - i \sqrt{19}) (3 - i \sqrt{19})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{3 (3 - i \sqrt{19}) - i \sqrt{19} (3 - i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Apply the distributive property.

$y = \frac{3 \cdot 3 + 3 (- i \sqrt{19}) - i \sqrt{19} (3 - i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Apply the distributive property.

$y = \frac{3 \cdot 3 + 3 (- i \sqrt{19}) - i \sqrt{19} \cdot 3 - i \sqrt{19} (- i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$y = \frac{9 + 3 (- i \sqrt{19}) - i \sqrt{19} \cdot 3 - i \sqrt{19} (- i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Multiply $- 1$ by $3$ .

$y = \frac{9 - 3 (i \sqrt{19}) - i \sqrt{19} \cdot 3 - i \sqrt{19} (- i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Multiply $3$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} - i \sqrt{19} (- i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Multiply $- i \sqrt{19} (- i \sqrt{19})$ .

Multiply $- 1$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 1 i \sqrt{19} (i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Multiply $\sqrt{19}$ by $1$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + \sqrt{19} i (i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Raise $\sqrt{19}$ to the power of $1$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{1} \sqrt{19} i i}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Raise $\sqrt{19}$ to the power of $1$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{1} {\sqrt{19}}^{1} i i}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{1 + 1} i i}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} i i}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} (i^{1} i)}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} (i^{1} i^{1})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} i^{1 + 1}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Add $1$ and $1$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} i^{2}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {(19^{\frac{1}{2}})}^{2} i^{2}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19^{\frac{1}{2} \cdot 2} i^{2}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19^{\frac{2}{2}} i^{2}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19^{\frac{2}{2}} i^{2}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Rewrite the expression.

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19^{1} i^{2}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Evaluate the exponent.

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19 i^{2}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

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

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19 \cdot - 1}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Multiply $19$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} - 19}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Subtract $19$ from $9$ .

$y = \frac{- 3 i \sqrt{19} - 3 i \sqrt{19} - 10}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Subtract $3 i \sqrt{19}$ from $- 3 i \sqrt{19}$ .

$y = \frac{- 6 i \sqrt{19} - 10}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Reorder $- 6 i \sqrt{19}$ and $- 10$ .

$y = \frac{- 10 - 6 i \sqrt{19}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Cancel the common factor of $- 10 - 6 i \sqrt{19}$ and $4$ .

Factor $2$ out of $- 10$ .

$y = \frac{2 (- 5) - 6 i \sqrt{19}}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Factor $2$ out of $- 6 i \sqrt{19}$ .

$y = \frac{2 (- 5) + 2 (- 3 i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Factor $2$ out of $2 (- 5) + 2 (- 3 i \sqrt{19})$ .

$y = \frac{2 (- 5 - 3 i \sqrt{19})}{4} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (- 5 - 3 i \sqrt{19})}{2 \cdot 2} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Cancel the common factor.

$y = \frac{2 (- 5 - 3 i \sqrt{19})}{2 \cdot 2} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Rewrite the expression.

$y = \frac{- 5 - 3 i \sqrt{19}}{2} - 2 \frac{3 - i \sqrt{19}}{2} - 3$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$y = \frac{- 5 - 3 i \sqrt{19}}{2} + 2 (- 1) \frac{3 - i \sqrt{19}}{2} - 3$

Cancel the common factor.

$y = \frac{- 5 - 3 i \sqrt{19}}{2} + 2 \cdot - 1 \frac{3 - i \sqrt{19}}{2} - 3$

Rewrite the expression.

$y = \frac{- 5 - 3 i \sqrt{19}}{2} - 1 (3 - i \sqrt{19}) - 3$

Apply the distributive property.

$y = \frac{- 5 - 3 i \sqrt{19}}{2} - 1 \cdot 3 - 1 (- i \sqrt{19}) - 3$

Multiply $- 1$ by $3$ .

$y = \frac{- 5 - 3 i \sqrt{19}}{2} - 3 - 1 (- i \sqrt{19}) - 3$

Multiply $- 1 (- i \sqrt{19})$ .

Multiply $- 1$ by $- 1$ .

$y = \frac{- 5 - 3 i \sqrt{19}}{2} - 3 + 1 (i \sqrt{19}) - 3$

Multiply $\sqrt{19}$ by $1$ .

$y = \frac{- 5 - 3 i \sqrt{19}}{2} - 3 + \sqrt{19} i - 3$

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

$y = \frac{- 5 - 3 i \sqrt{19}}{2} - 3 \cdot \frac{2}{2} + \sqrt{19} i - 3$

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

$y = \frac{- 5 - 3 i \sqrt{19}}{2} + \frac{- 3 \cdot 2}{2} + \sqrt{19} i - 3$

Combine the numerators over the common denominator.

$y = \frac{- 11 - 3 i \sqrt{19}}{2} + \sqrt{19} i - 3$

To write $\sqrt{19} i$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 11 - 3 i \sqrt{19}}{2} + \sqrt{19} i \cdot \frac{2}{2} - 3$

Combine $\sqrt{19} i$ and $\frac{2}{2}$ .

$y = \frac{- 11 - 3 i \sqrt{19}}{2} + \frac{\sqrt{19} i \cdot 2}{2} - 3$

Combine the numerators over the common denominator.

$y = \frac{- 11 - i \sqrt{19}}{2} - 3$

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

$y = \frac{- 11 - i \sqrt{19}}{2} - 3 \cdot \frac{2}{2}$

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

$y = \frac{- 11 - i \sqrt{19}}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 17 - i \sqrt{19}}{2}$

Rewrite $- 17$ as $- 1 (17)$ .

$y = \frac{- 1 (17) - i \sqrt{19}}{2}$

Factor $- 1$ out of $- i \sqrt{19}$ .

$y = \frac{- 1 (17) - (i \sqrt{19})}{2}$

Factor $- 1$ out of $- 1 (17) - (i \sqrt{19})$ .

$y = \frac{- 1 (17 + i \sqrt{19})}{2}$

Move the negative in front of the fraction.

$y = - \frac{17 + i \sqrt{19}}{2}$

Step 5

List all of the solutions.

$y = - \frac{17 - i \sqrt{19}}{2}, x = \frac{3 + i \sqrt{19}}{2}$

$y = - \frac{17 + i \sqrt{19}}{2}, x = \frac{3 - i \sqrt{19}}{2}$

Step 6

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

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