Solve the System of Equations y-15=x^2+4x x-y=1

$y - 15 = x^{2} + 4 x$ $x - y = 1$

Step 1

Add $15$ to both sides of the equation.

$y = x^{2} + 4 x + 15$

$x - y = 1$

Step 2

Replace all occurrences of $y$ with $x^{2} + 4 x + 15$ in each equation.

Replace all occurrences of $y$ in $x - y = 1$ with $x^{2} + 4 x + 15$ .

$x - (x^{2} + 4 x + 15) = 1$

$y = x^{2} + 4 x + 15$

Simplify the left side.

Simplify $x - (x^{2} + 4 x + 15)$ .

Simplify each term.

Apply the distributive property.

$x - x^{2} - (4 x) - 1 \cdot 15 = 1$

$y = x^{2} + 4 x + 15$

Simplify.

Multiply $4$ by $- 1$ .

$x - x^{2} - 4 x - 1 \cdot 15 = 1$

$y = x^{2} + 4 x + 15$

Multiply $- 1$ by $15$ .

$x - x^{2} - 4 x - 15 = 1$

$y = x^{2} + 4 x + 15$

$x - x^{2} - 4 x - 15 = 1$

$y = x^{2} + 4 x + 15$

$x - x^{2} - 4 x - 15 = 1$

$y = x^{2} + 4 x + 15$

Subtract $4 x$ from $x$ .

$- x^{2} - 3 x - 15 = 1$

$y = x^{2} + 4 x + 15$

$- x^{2} - 3 x - 15 = 1$

$y = x^{2} + 4 x + 15$

$- x^{2} - 3 x - 15 = 1$

$y = x^{2} + 4 x + 15$

$- x^{2} - 3 x - 15 = 1$

$y = x^{2} + 4 x + 15$

Step 3

Solve for $x$ in $- x^{2} - 3 x - 15 = 1$ .

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

Subtract $1$ from both sides of the equation.

$- x^{2} - 3 x - 15 - 1 = 0$

$y = x^{2} + 4 x + 15$

Subtract $1$ from $- 15$ .

$- x^{2} - 3 x - 16 = 0$

$y = x^{2} + 4 x + 15$

$- x^{2} - 3 x - 16 = 0$

$y = x^{2} + 4 x + 15$

Use the quadratic formula to find the solutions.

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

$y = x^{2} + 4 x + 15$

Substitute the values $a = - 1$ , $b = - 3$ , and $c = - 16$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (- 1 \cdot - 16)}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Simplify.

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 16}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $- 4 \cdot - 1 \cdot - 16$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4 \cdot - 16}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $4$ by $- 16$ .

$x = \frac{3 \pm \sqrt{9 - 64}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

$x = \frac{3 \pm \sqrt{9 - 64}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Subtract $64$ from $9$ .

$x = \frac{3 \pm \sqrt{- 55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm \sqrt{- 1 \cdot 55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm i \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

$x = \frac{3 \pm i \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $2$ by $- 1$ .

$x = \frac{3 \pm i \sqrt{55}}{- 2}$

$y = x^{2} + 4 x + 15$

Move the negative in front of the fraction.

$x = - \frac{3 \pm i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

$x = - \frac{3 \pm i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

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

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 16}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $- 4 \cdot - 1 \cdot - 16$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4 \cdot - 16}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $4$ by $- 16$ .

$x = \frac{3 \pm \sqrt{9 - 64}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

$x = \frac{3 \pm \sqrt{9 - 64}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Subtract $64$ from $9$ .

$x = \frac{3 \pm \sqrt{- 55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm \sqrt{- 1 \cdot 55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm i \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

$x = \frac{3 \pm i \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $2$ by $- 1$ .

$x = \frac{3 \pm i \sqrt{55}}{- 2}$

$y = x^{2} + 4 x + 15$

Move the negative in front of the fraction.

$x = - \frac{3 \pm i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

Change the $\pm$ to $+$ .

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

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

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1 \cdot - 16}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $- 4 \cdot - 1 \cdot - 16$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4 \cdot - 16}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $4$ by $- 16$ .

$x = \frac{3 \pm \sqrt{9 - 64}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

$x = \frac{3 \pm \sqrt{9 - 64}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Subtract $64$ from $9$ .

$x = \frac{3 \pm \sqrt{- 55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm \sqrt{- 1 \cdot 55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

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

$x = \frac{3 \pm i \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

$x = \frac{3 \pm i \sqrt{55}}{2 \cdot - 1}$

$y = x^{2} + 4 x + 15$

Multiply $2$ by $- 1$ .

$x = \frac{3 \pm i \sqrt{55}}{- 2}$

$y = x^{2} + 4 x + 15$

Move the negative in front of the fraction.

$x = - \frac{3 \pm i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

Change the $\pm$ to $-$ .

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

The final answer is the combination of both solutions.

$x = - \frac{3 + i \sqrt{55}}{2}, - \frac{3 - i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

$x = - \frac{3 + i \sqrt{55}}{2}, - \frac{3 - i \sqrt{55}}{2}$

$y = x^{2} + 4 x + 15$

Step 4

Replace all occurrences of $x$ with $- \frac{3 + i \sqrt{55}}{2}$ in each equation.

Replace all occurrences of $x$ in $y = x^{2} + 4 x + 15$ with $- \frac{3 + i \sqrt{55}}{2}$ .

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

$x = - \frac{3 + i \sqrt{55}}{2}$

Simplify the right side.

Simplify ${(- \frac{3 + i \sqrt{55}}{2})}^{2} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$ .

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{3 + i \sqrt{55}}{2}$ .

$y = {(- 1)}^{2} {(\frac{3 + i \sqrt{55}}{2})}^{2} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Apply the product rule to $\frac{3 + i \sqrt{55}}{2}$ .

$y = {(- 1)}^{2} (\frac{{(3 + i \sqrt{55})}^{2}}{2^{2}}) + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = {(- 1)}^{2} (\frac{{(3 + i \sqrt{55})}^{2}}{2^{2}}) + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = 1 (\frac{{(3 + i \sqrt{55})}^{2}}{2^{2}}) + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Multiply $\frac{{(3 + i \sqrt{55})}^{2}}{2^{2}}$ by $1$ .

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

$x = - \frac{3 + i \sqrt{55}}{2}$

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

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

$x = - \frac{3 + i \sqrt{55}}{2}$

Rewrite ${(3 + i \sqrt{55})}^{2}$ as $(3 + i \sqrt{55}) (3 + i \sqrt{55})$ .

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

$x = - \frac{3 + i \sqrt{55}}{2}$

Expand $(3 + i \sqrt{55}) (3 + i \sqrt{55})$ using the FOIL Method.

Apply the distributive property.

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

$x = - \frac{3 + i \sqrt{55}}{2}$

Apply the distributive property.

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

$x = - \frac{3 + i \sqrt{55}}{2}$

Apply the distributive property.

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

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$x = - \frac{3 + i \sqrt{55}}{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$y = \frac{9 + 3 (i \sqrt{55}) + i \sqrt{55} \cdot 3 + i \sqrt{55} (i \sqrt{55})}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Move $3$ to the left of $i \sqrt{55}$ .

$y = \frac{9 + 3 i \sqrt{55} + 3 \cdot (i \sqrt{55}) + i \sqrt{55} (i \sqrt{55})}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i i \sqrt{55} \sqrt{55}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i i \sqrt{55} \sqrt{55}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i^{1 + 1} \sqrt{55} \sqrt{55}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Add $1$ and $1$ .

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i^{2} \sqrt{55} \sqrt{55}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i^{2} (\sqrt{55} \sqrt{55})}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i^{2} (\sqrt{55} \sqrt{55})}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i^{2} {\sqrt{55}}^{1 + 1}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Add $1$ and $1$ .

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i^{2} {\sqrt{55}}^{2}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} + i^{2} {\sqrt{55}}^{2}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 {\sqrt{55}}^{2}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 {(55^{\frac{1}{2}})}^{2}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 \cdot 55^{\frac{1}{2} \cdot 2}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 \cdot 55^{\frac{2}{2}}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 \cdot 55^{\frac{2}{2}}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Rewrite the expression.

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 \cdot 55}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 \cdot 55}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Evaluate the exponent.

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 \cdot 55}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 1 \cdot 55}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Multiply $- 1$ by $55$ .

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 55}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{9 + 3 i \sqrt{55} + 3 i \sqrt{55} - 55}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Subtract $55$ from $9$ .

$y = \frac{3 i \sqrt{55} + 3 i \sqrt{55} - 46}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Add $3 i \sqrt{55}$ and $3 i \sqrt{55}$ .

$y = \frac{6 i \sqrt{55} - 46}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{6 i \sqrt{55} - 46}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Reorder $6 i \sqrt{55}$ and $- 46$ .

$y = \frac{- 46 + 6 i \sqrt{55}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Cancel the common factor of $- 46 + 6 i \sqrt{55}$ and $4$ .

Factor $2$ out of $- 46$ .

$y = \frac{2 (- 23) + 6 i \sqrt{55}}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Factor $2$ out of $6 i \sqrt{55}$ .

$y = \frac{2 (- 23) + 2 (3 i \sqrt{55})}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Factor $2$ out of $2 (- 23) + 2 (3 i \sqrt{55})$ .

$y = \frac{2 (- 23 + 3 i \sqrt{55})}{4} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (- 23 + 3 i \sqrt{55})}{2 \cdot 2} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Cancel the common factor.

$y = \frac{2 (- 23 + 3 i \sqrt{55})}{2 \cdot 2} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Rewrite the expression.

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + 4 (- \frac{3 + i \sqrt{55}}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3 + i \sqrt{55}}{2}$ into the numerator.

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + 4 (\frac{- (3 + i \sqrt{55})}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Factor $2$ out of $4$ .

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + 2 (2) (\frac{- (3 + i \sqrt{55})}{2}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Cancel the common factor.

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + 2 \cdot (2 (\frac{- (3 + i \sqrt{55})}{2})) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Rewrite the expression.

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + 2 (- (3 + i \sqrt{55})) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + 2 (- (3 + i \sqrt{55})) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Multiply $- 1$ by $2$ .

$y = \frac{- 23 + 3 i \sqrt{55}}{2} - 2 (3 + i \sqrt{55}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Apply the distributive property.

$y = \frac{- 23 + 3 i \sqrt{55}}{2} - 2 \cdot 3 - 2 (i \sqrt{55}) + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Multiply $- 2$ by $3$ .

$y = \frac{- 23 + 3 i \sqrt{55}}{2} - 6 - 2 i \sqrt{55} + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = \frac{- 23 + 3 i \sqrt{55}}{2} - 6 - 2 i \sqrt{55} + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 23 + 3 i \sqrt{55}}{2} - 6 \cdot \frac{2}{2} - 2 i \sqrt{55} + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Combine $- 6$ and $\frac{2}{2}$ .

$y = \frac{- 23 + 3 i \sqrt{55}}{2} + \frac{- 6 \cdot 2}{2} - 2 i \sqrt{55} + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 35 + 3 i \sqrt{55}}{2} - 2 i \sqrt{55} + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

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

$y = \frac{- 35 + 3 i \sqrt{55}}{2} - 2 i \sqrt{55} \cdot \frac{2}{2} + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Combine $- 2 i \sqrt{55}$ and $\frac{2}{2}$ .

$y = \frac{- 35 + 3 i \sqrt{55}}{2} + \frac{- 2 i \sqrt{55} \cdot 2}{2} + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 35 - i \sqrt{55}}{2} + 15$

$x = - \frac{3 + i \sqrt{55}}{2}$

To write $15$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 35 - i \sqrt{55}}{2} + 15 \cdot \frac{2}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

Combine $15$ and $\frac{2}{2}$ .

$y = \frac{- 35 - i \sqrt{55}}{2} + \frac{15 \cdot 2}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 5 - i \sqrt{55}}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

Rewrite $- 5$ as $- 1 (5)$ .

$y = \frac{- 1 \cdot 5 - i \sqrt{55}}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

Factor $- 1$ out of $- i \sqrt{55}$ .

$y = \frac{- 1 \cdot 5 - (i \sqrt{55})}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

Factor $- 1$ out of $- 1 (5) - (i \sqrt{55})$ .

$y = \frac{- 1 (5 + i \sqrt{55})}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

Move the negative in front of the fraction.

$y = - \frac{5 + i \sqrt{55}}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = - \frac{5 + i \sqrt{55}}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = - \frac{5 + i \sqrt{55}}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

$y = - \frac{5 + i \sqrt{55}}{2}$

$x = - \frac{3 + i \sqrt{55}}{2}$

Step 5

Replace all occurrences of $x$ with $- \frac{3 - i \sqrt{55}}{2}$ in each equation.

Replace all occurrences of $x$ in $y = x^{2} + 4 x + 15$ with $- \frac{3 - i \sqrt{55}}{2}$ .

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

$x = - \frac{3 - i \sqrt{55}}{2}$

Simplify the right side.

Simplify ${(- \frac{3 - i \sqrt{55}}{2})}^{2} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$ .

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{3 - i \sqrt{55}}{2}$ .

$y = {(- 1)}^{2} {(\frac{3 - i \sqrt{55}}{2})}^{2} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Apply the product rule to $\frac{3 - i \sqrt{55}}{2}$ .

$y = {(- 1)}^{2} (\frac{{(3 - i \sqrt{55})}^{2}}{2^{2}}) + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = {(- 1)}^{2} (\frac{{(3 - i \sqrt{55})}^{2}}{2^{2}}) + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = 1 (\frac{{(3 - i \sqrt{55})}^{2}}{2^{2}}) + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Multiply $\frac{{(3 - i \sqrt{55})}^{2}}{2^{2}}$ by $1$ .

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

$x = - \frac{3 - i \sqrt{55}}{2}$

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

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

$x = - \frac{3 - i \sqrt{55}}{2}$

Rewrite ${(3 - i \sqrt{55})}^{2}$ as $(3 - i \sqrt{55}) (3 - i \sqrt{55})$ .

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

$x = - \frac{3 - i \sqrt{55}}{2}$

Expand $(3 - i \sqrt{55}) (3 - i \sqrt{55})$ using the FOIL Method.

Apply the distributive property.

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

$x = - \frac{3 - i \sqrt{55}}{2}$

Apply the distributive property.

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

$x = - \frac{3 - i \sqrt{55}}{2}$

Apply the distributive property.

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

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$x = - \frac{3 - i \sqrt{55}}{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$y = \frac{9 + 3 (- i \sqrt{55}) - i \sqrt{55} \cdot 3 - i \sqrt{55} (- i \sqrt{55})}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Multiply $- 1$ by $3$ .

$y = \frac{9 - 3 (i \sqrt{55}) - i \sqrt{55} \cdot 3 - i \sqrt{55} (- i \sqrt{55})}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Multiply $3$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} - i \sqrt{55} (- i \sqrt{55})}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

Multiply $- 1$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 1 i \sqrt{55} (i \sqrt{55})}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + \sqrt{55} i (i \sqrt{55})}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + \sqrt{55} \sqrt{55} i i}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + \sqrt{55} \sqrt{55} i i}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + {\sqrt{55}}^{1 + 1} i i}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Add $1$ and $1$ .

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + {\sqrt{55}}^{2} i i}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + {\sqrt{55}}^{2} (i i)}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + {\sqrt{55}}^{2} (i i)}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + {\sqrt{55}}^{2} i^{1 + 1}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Add $1$ and $1$ .

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + {\sqrt{55}}^{2} i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + {\sqrt{55}}^{2} i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + {(55^{\frac{1}{2}})}^{2} i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 55^{\frac{1}{2} \cdot 2} i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 55^{\frac{2}{2}} i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 55^{\frac{2}{2}} i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Rewrite the expression.

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 55 i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 55 i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Evaluate the exponent.

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 55 i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 55 i^{2}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

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

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} + 55 \cdot - 1}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Multiply $55$ by $- 1$ .

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} - 55}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{9 - 3 i \sqrt{55} - 3 i \sqrt{55} - 55}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Subtract $55$ from $9$ .

$y = \frac{- 3 i \sqrt{55} - 3 i \sqrt{55} - 46}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Subtract $3 i \sqrt{55}$ from $- 3 i \sqrt{55}$ .

$y = \frac{- 6 i \sqrt{55} - 46}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{- 6 i \sqrt{55} - 46}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Reorder $- 6 i \sqrt{55}$ and $- 46$ .

$y = \frac{- 46 - 6 i \sqrt{55}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Cancel the common factor of $- 46 - 6 i \sqrt{55}$ and $4$ .

Factor $2$ out of $- 46$ .

$y = \frac{2 (- 23) - 6 i \sqrt{55}}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Factor $2$ out of $- 6 i \sqrt{55}$ .

$y = \frac{2 (- 23) + 2 (- 3 i \sqrt{55})}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Factor $2$ out of $2 (- 23) + 2 (- 3 i \sqrt{55})$ .

$y = \frac{2 (- 23 - 3 i \sqrt{55})}{4} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{2 (- 23 - 3 i \sqrt{55})}{2 \cdot 2} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Cancel the common factor.

$y = \frac{2 (- 23 - 3 i \sqrt{55})}{2 \cdot 2} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Rewrite the expression.

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + 4 (- \frac{3 - i \sqrt{55}}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3 - i \sqrt{55}}{2}$ into the numerator.

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + 4 (\frac{- (3 - i \sqrt{55})}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Factor $2$ out of $4$ .

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + 2 (2) (\frac{- (3 - i \sqrt{55})}{2}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Cancel the common factor.

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + 2 \cdot (2 (\frac{- (3 - i \sqrt{55})}{2})) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Rewrite the expression.

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + 2 (- (3 - i \sqrt{55})) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + 2 (- (3 - i \sqrt{55})) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Multiply $- 1$ by $2$ .

$y = \frac{- 23 - 3 i \sqrt{55}}{2} - 2 (3 - i \sqrt{55}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Apply the distributive property.

$y = \frac{- 23 - 3 i \sqrt{55}}{2} - 2 \cdot 3 - 2 (- i \sqrt{55}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Multiply $- 2$ by $3$ .

$y = \frac{- 23 - 3 i \sqrt{55}}{2} - 6 - 2 (- i \sqrt{55}) + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Multiply $- 1$ by $- 2$ .

$y = \frac{- 23 - 3 i \sqrt{55}}{2} - 6 + 2 i \sqrt{55} + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = \frac{- 23 - 3 i \sqrt{55}}{2} - 6 + 2 i \sqrt{55} + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 23 - 3 i \sqrt{55}}{2} - 6 \cdot \frac{2}{2} + 2 i \sqrt{55} + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Combine $- 6$ and $\frac{2}{2}$ .

$y = \frac{- 23 - 3 i \sqrt{55}}{2} + \frac{- 6 \cdot 2}{2} + 2 i \sqrt{55} + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 35 - 3 i \sqrt{55}}{2} + 2 i \sqrt{55} + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

To write $2 i \sqrt{55}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 35 - 3 i \sqrt{55}}{2} + 2 i \sqrt{55} \cdot \frac{2}{2} + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Combine $2 i \sqrt{55}$ and $\frac{2}{2}$ .

$y = \frac{- 35 - 3 i \sqrt{55}}{2} + \frac{2 i \sqrt{55} \cdot 2}{2} + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 35 + i \sqrt{55}}{2} + 15$

$x = - \frac{3 - i \sqrt{55}}{2}$

To write $15$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 35 + i \sqrt{55}}{2} + 15 \cdot \frac{2}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

Combine $15$ and $\frac{2}{2}$ .

$y = \frac{- 35 + i \sqrt{55}}{2} + \frac{15 \cdot 2}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 5 + i \sqrt{55}}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

Rewrite $- 5$ as $- 1 (5)$ .

$y = \frac{- 1 \cdot 5 + i \sqrt{55}}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

Factor $- 1$ out of $i \sqrt{55}$ .

$y = \frac{- 1 \cdot 5 - (- i \sqrt{55})}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

Factor $- 1$ out of $- 1 (5) - (- i \sqrt{55})$ .

$y = \frac{- 1 (5 - i \sqrt{55})}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

Move the negative in front of the fraction.

$y = - \frac{5 - i \sqrt{55}}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = - \frac{5 - i \sqrt{55}}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = - \frac{5 - i \sqrt{55}}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

$y = - \frac{5 - i \sqrt{55}}{2}$

$x = - \frac{3 - i \sqrt{55}}{2}$

Step 6

List all of the solutions.

$y = - \frac{5 + i \sqrt{55}}{2}, x = - \frac{3 + i \sqrt{55}}{2}$

$y = - \frac{5 - i \sqrt{55}}{2}, x = - \frac{3 - i \sqrt{55}}{2}$

Step 7

Solve the System of Equations y-15=x^2+4x x-y=1

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