Algebra Examples

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Add $2 x$ to both sides of the equation.

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

Step 2.2.2

Add $- 5 x$ and $2 x$ .

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

Step 2.3

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

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Step 2.3.1

Subtract $5$ from both sides of the equation.

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

Step 2.3.2

Subtract $5$ from $- 2$ .

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

Step 2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = - x^{2} - 5 x - 2$

Step 2.5

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

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (- 1 \cdot - 7)}}{2 \cdot - 1} + y = - x^{2} - 5 x - 2$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

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

Step 2.6.1.2

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

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Step 2.6.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 2.6.1.2.2

Multiply $4$ by $- 7$ .

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

Step 2.6.1.3

Subtract $28$ from $9$ .

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

Step 2.6.1.4

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

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

Step 2.6.1.5

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

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

Step 2.6.1.6

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

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

Step 2.6.2

Multiply $2$ by $- 1$ .

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

Step 2.6.3

Move the negative in front of the fraction.

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

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

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

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

Step 2.7.1.2

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

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Step 2.7.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 2.7.1.2.2

Multiply $4$ by $- 7$ .

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

Step 2.7.1.3

Subtract $28$ from $9$ .

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

Step 2.7.1.4

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

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

Step 2.7.1.5

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

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

Step 2.7.1.6

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

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

Step 2.7.2

Multiply $2$ by $- 1$ .

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

Step 2.7.3

Move the negative in front of the fraction.

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

Step 2.7.4

Change the $\pm$ to $+$ .

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

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

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

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

Step 2.8.1.2

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

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Step 2.8.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 2.8.1.2.2

Multiply $4$ by $- 7$ .

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

Step 2.8.1.3

Subtract $28$ from $9$ .

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

Step 2.8.1.4

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

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

Step 2.8.1.5

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

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

Step 2.8.1.6

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

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

Step 2.8.2

Multiply $2$ by $- 1$ .

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

Step 2.8.3

Move the negative in front of the fraction.

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

Step 2.8.4

Change the $\pm$ to $-$ .

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

Step 2.9

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

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Step 3.1

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

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

Step 3.2

Simplify $- {(- \frac{3 + i \sqrt{19}}{2})}^{2} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

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

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

Step 3.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.2.1.2.1

Move ${(- 1)}^{2}$ .

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

Step 3.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 3.2.1.2.2.1

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

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

Step 3.2.1.2.2.2

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

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

Step 3.2.1.2.3

Add $2$ and $1$ .

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

Step 3.2.1.3

Raise $- 1$ to the power of $3$ .

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

Step 3.2.1.4

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

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

Step 3.2.1.5

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

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

Step 3.2.1.6

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

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Step 3.2.1.6.1

Apply the distributive property.

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

Step 3.2.1.6.2

Apply the distributive property.

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

Step 3.2.1.6.3

Apply the distributive property.

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

Step 3.2.1.7

Simplify and combine like terms.

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Step 3.2.1.7.1

Simplify each term.

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Step 3.2.1.7.1.1

Multiply $3$ by $3$ .

$y = - \frac{9 + 3 (i \sqrt{19}) + i \sqrt{19} \cdot 3 + i \sqrt{19} (i \sqrt{19})}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.2

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 \cdot (i \sqrt{19}) + i \sqrt{19} (i \sqrt{19})}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.3

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

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Step 3.2.1.7.1.3.1

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{1} i \sqrt{19} \sqrt{19}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.3.2

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{1} i^{1} \sqrt{19} \sqrt{19}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.3.3

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{1 + 1} \sqrt{19} \sqrt{19}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.3.4

Add $1$ and $1$ .

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} \sqrt{19} \sqrt{19}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.3.5

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} ({\sqrt{19}}^{1} \sqrt{19})}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.3.6

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} ({\sqrt{19}}^{1} {\sqrt{19}}^{1})}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.3.7

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} {\sqrt{19}}^{1 + 1}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.3.8

Add $1$ and $1$ .

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} + i^{2} {\sqrt{19}}^{2}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.4

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 {\sqrt{19}}^{2}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.5

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

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Step 3.2.1.7.1.5.1

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 {(19^{\frac{1}{2}})}^{2}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.5.2

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19^{\frac{1}{2} \cdot 2}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.5.3

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

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19^{\frac{2}{2}}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 3.2.1.7.1.5.4.1

Cancel the common factor.

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19^{\frac{2}{2}}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.5.4.2

Rewrite the expression.

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19^{1}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.5.5

Evaluate the exponent.

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 1 \cdot 19}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.1.6

Multiply $- 1$ by $19$ .

$y = - \frac{9 + 3 i \sqrt{19} + 3 i \sqrt{19} - 19}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.2

Subtract $19$ from $9$ .

$y = - \frac{3 i \sqrt{19} + 3 i \sqrt{19} - 10}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.7.3

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

$y = - \frac{6 i \sqrt{19} - 10}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.8

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

$y = - \frac{- 10 + 6 i \sqrt{19}}{4} - 5 (- \frac{3 + i \sqrt{19}}{2}) - 2$

Step 3.2.1.9

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

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Step 3.2.1.9.1

Factor $2$ out of $- 10$ .

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

Step 3.2.1.9.2

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

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

Step 3.2.1.9.3

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

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

Step 3.2.1.9.4

Cancel the common factors.

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Step 3.2.1.9.4.1

Factor $2$ out of $4$ .

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

Step 3.2.1.9.4.2

Cancel the common factor.

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

Step 3.2.1.9.4.3

Rewrite the expression.

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

Step 3.2.1.10

Multiply $- 5 (- \frac{3 + i \sqrt{19}}{2})$ .

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Step 3.2.1.10.1

Multiply $- 1$ by $- 5$ .

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

Step 3.2.1.10.2

Combine $5$ and $\frac{3 + i \sqrt{19}}{2}$ .

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

Step 3.2.2

Combine the numerators over the common denominator.

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

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

Apply the distributive property.

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

Step 3.2.3.2

Multiply $- 1$ by $- 5$ .

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

Step 3.2.3.3

Multiply $3$ by $- 1$ .

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

Step 3.2.3.4

Apply the distributive property.

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

Step 3.2.3.5

Multiply $5$ by $3$ .

$y = - 2 + \frac{5 - 3 i \sqrt{19} + 15 + 5 i \sqrt{19}}{2}$

Step 3.2.4

Simplify terms.

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Step 3.2.4.1

Add $5$ and $15$ .

$y = - 2 + \frac{- 3 i \sqrt{19} + 20 + 5 i \sqrt{19}}{2}$

Step 3.2.4.2

Add $- 3 i \sqrt{19}$ and $5 i \sqrt{19}$ .

$y = - 2 + \frac{2 i \sqrt{19} + 20}{2}$

Step 3.2.4.3

Reorder $2 i \sqrt{19}$ and $20$ .

$y = - 2 + \frac{20 + 2 i \sqrt{19}}{2}$

Step 3.2.4.4

Cancel the common factor of $20 + 2 i \sqrt{19}$ and $2$ .

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Step 3.2.4.4.1

Factor $2$ out of $20$ .

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

Step 3.2.4.4.2

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

$y = - 2 + \frac{2 \cdot 10 + 2 (i \sqrt{19})}{2}$

Step 3.2.4.4.3

Factor $2$ out of $2 (10) + 2 (i \sqrt{19})$ .

$y = - 2 + \frac{2 (10 + i \sqrt{19})}{2}$

Step 3.2.4.4.4

Cancel the common factors.

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Step 3.2.4.4.4.1

Factor $2$ out of $2$ .

$y = - 2 + \frac{2 (10 + i \sqrt{19})}{2 (1)}$

Step 3.2.4.4.4.2

Cancel the common factor.

$y = - 2 + \frac{2 (10 + i \sqrt{19})}{2 \cdot 1}$

Step 3.2.4.4.4.3

Rewrite the expression.

$y = - 2 + \frac{10 + i \sqrt{19}}{1}$

Step 3.2.4.4.4.4

Divide $10 + i \sqrt{19}$ by $1$ .

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

Step 3.2.4.5

Add $- 2$ and $10$ .

$y = 8 + i \sqrt{19}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify $- {(- \frac{3 - i \sqrt{19}}{2})}^{2} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

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

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

Step 4.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.2.1.2.1

Move ${(- 1)}^{2}$ .

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

Step 4.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 4.2.1.2.2.1

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

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

Step 4.2.1.2.2.2

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

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

Step 4.2.1.2.3

Add $2$ and $1$ .

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

Step 4.2.1.3

Raise $- 1$ to the power of $3$ .

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

Step 4.2.1.4

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

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

Step 4.2.1.5

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

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

Step 4.2.1.6

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

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Step 4.2.1.6.1

Apply the distributive property.

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

Step 4.2.1.6.2

Apply the distributive property.

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

Step 4.2.1.6.3

Apply the distributive property.

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

Step 4.2.1.7

Simplify and combine like terms.

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Step 4.2.1.7.1

Simplify each term.

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Step 4.2.1.7.1.1

Multiply $3$ by $3$ .

$y = - \frac{9 + 3 (- i \sqrt{19}) - i \sqrt{19} \cdot 3 - i \sqrt{19} (- i \sqrt{19})}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.2

Multiply $- 1$ by $3$ .

$y = - \frac{9 - 3 (i \sqrt{19}) - i \sqrt{19} \cdot 3 - i \sqrt{19} (- i \sqrt{19})}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.3

Multiply $3$ by $- 1$ .

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} - i \sqrt{19} (- i \sqrt{19})}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4

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

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Step 4.2.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 1 i \sqrt{19} (i \sqrt{19})}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.2

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + \sqrt{19} i (i \sqrt{19})}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.3

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{1} \sqrt{19} i i}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.4

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{1} {\sqrt{19}}^{1} i i}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.5

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{1 + 1} i i}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.6

Add $1$ and $1$ .

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} i i}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.7

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} (i^{1} i)}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.8

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} (i^{1} i^{1})}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.9

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} i^{1 + 1}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.4.10

Add $1$ and $1$ .

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {\sqrt{19}}^{2} i^{2}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.5

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

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Step 4.2.1.7.1.5.1

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + {(19^{\frac{1}{2}})}^{2} i^{2}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.5.2

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19^{\frac{1}{2} \cdot 2} i^{2}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.5.3

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19^{\frac{2}{2}} i^{2}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.1.7.1.5.4.1

Cancel the common factor.

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19^{\frac{2}{2}} i^{2}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.5.4.2

Rewrite the expression.

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19^{1} i^{2}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.5.5

Evaluate the exponent.

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19 i^{2}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.6

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

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} + 19 \cdot - 1}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.1.7

Multiply $19$ by $- 1$ .

$y = - \frac{9 - 3 i \sqrt{19} - 3 i \sqrt{19} - 19}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.2

Subtract $19$ from $9$ .

$y = - \frac{- 3 i \sqrt{19} - 3 i \sqrt{19} - 10}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.7.3

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

$y = - \frac{- 6 i \sqrt{19} - 10}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.8

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

$y = - \frac{- 10 - 6 i \sqrt{19}}{4} - 5 (- \frac{3 - i \sqrt{19}}{2}) - 2$

Step 4.2.1.9

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

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Step 4.2.1.9.1

Factor $2$ out of $- 10$ .

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

Step 4.2.1.9.2

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

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

Step 4.2.1.9.3

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

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

Step 4.2.1.9.4

Cancel the common factors.

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Step 4.2.1.9.4.1

Factor $2$ out of $4$ .

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

Step 4.2.1.9.4.2

Cancel the common factor.

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

Step 4.2.1.9.4.3

Rewrite the expression.

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

Step 4.2.1.10

Multiply $- 5 (- \frac{3 - i \sqrt{19}}{2})$ .

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Step 4.2.1.10.1

Multiply $- 1$ by $- 5$ .

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

Step 4.2.1.10.2

Combine $5$ and $\frac{3 - i \sqrt{19}}{2}$ .

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

Step 4.2.2

Combine the numerators over the common denominator.

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Apply the distributive property.

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

Step 4.2.3.2

Multiply $- 1$ by $- 5$ .

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

Step 4.2.3.3

Multiply $- 3$ by $- 1$ .

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

Step 4.2.3.4

Apply the distributive property.

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

Step 4.2.3.5

Multiply $5$ by $3$ .

$y = - 2 + \frac{5 + 3 i \sqrt{19} + 15 + 5 (- i \sqrt{19})}{2}$

Step 4.2.3.6

Multiply $- 1$ by $5$ .

$y = - 2 + \frac{5 + 3 i \sqrt{19} + 15 - 5 i \sqrt{19}}{2}$

Step 4.2.4

Simplify terms.

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Step 4.2.4.1

Add $5$ and $15$ .

$y = - 2 + \frac{3 i \sqrt{19} + 20 - 5 i \sqrt{19}}{2}$

Step 4.2.4.2

Subtract $5 i \sqrt{19}$ from $3 i \sqrt{19}$ .

$y = - 2 + \frac{- 2 i \sqrt{19} + 20}{2}$

Step 4.2.4.3

Reorder $- 2 i \sqrt{19}$ and $20$ .

$y = - 2 + \frac{20 - 2 i \sqrt{19}}{2}$

Step 4.2.4.4

Cancel the common factor of $20 - 2 i \sqrt{19}$ and $2$ .

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Step 4.2.4.4.1

Factor $2$ out of $20$ .

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

Step 4.2.4.4.2

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

$y = - 2 + \frac{2 \cdot 10 + 2 (- i \sqrt{19})}{2}$

Step 4.2.4.4.3

Factor $2$ out of $2 (10) + 2 (- i \sqrt{19})$ .

$y = - 2 + \frac{2 (10 - i \sqrt{19})}{2}$

Step 4.2.4.4.4

Cancel the common factors.

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Step 4.2.4.4.4.1

Factor $2$ out of $2$ .

$y = - 2 + \frac{2 (10 - i \sqrt{19})}{2 (1)}$

Step 4.2.4.4.4.2

Cancel the common factor.

$y = - 2 + \frac{2 (10 - i \sqrt{19})}{2 \cdot 1}$

Step 4.2.4.4.4.3

Rewrite the expression.

$y = - 2 + \frac{10 - i \sqrt{19}}{1}$

Step 4.2.4.4.4.4

Divide $10 - i \sqrt{19}$ by $1$ .

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

Step 4.2.4.5

Add $- 2$ and $10$ .

$y = 8 - i \sqrt{19}$

Step 5

List all of the solutions.

$y = 8 + i \sqrt{19}, x = - \frac{3 + i \sqrt{19}}{2}$

$y = 8 - i \sqrt{19}, x = - \frac{3 - i \sqrt{19}}{2}$

Step 6