Algebra Examples

$y = x - 7$ $y = x^{2} + 2 x - 4$

Step 1

Eliminate the equal sides of each equation and combine.

$x - 7 = x^{2} + 2 x - 4$

Step 2

Solve $x - 7 = x^{2} + 2 x - 4$ 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.

$x^{2} + 2 x - 4 = x - 7$

Step 2.2

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

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Step 2.2.1

Subtract $x$ from both sides of the equation.

$x^{2} + 2 x - 4 - x = - 7$

Step 2.2.2

Subtract $x$ from $2 x$ .

$x^{2} + x - 4 = - 7$

Step 2.3

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

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Step 2.3.1

Add $7$ to both sides of the equation.

$x^{2} + x - 4 + 7 = 0$

Step 2.3.2

Add $- 4$ and $7$ .

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

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

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

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

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

One to any power is one.

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

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.6.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12}}{2 \cdot 1}$

Step 2.6.1.3

Subtract $12$ from $1$ .

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

Step 2.6.1.4

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

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

Step 2.6.1.5

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

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

Step 2.6.1.6

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

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

Step 2.6.2

Multiply $2$ by $1$ .

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

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

One to any power is one.

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

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.7.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12}}{2 \cdot 1}$

Step 2.7.1.3

Subtract $12$ from $1$ .

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

Step 2.7.1.4

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

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

Step 2.7.1.5

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

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

Step 2.7.1.6

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

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

Step 2.7.2

Multiply $2$ by $1$ .

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

Step 2.7.3

Change the $\pm$ to $+$ .

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

Step 2.7.4

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

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

Step 2.7.5

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

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

Step 2.7.6

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

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

Step 2.7.7

Move the negative in front of the fraction.

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

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

One to any power is one.

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

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.8.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 - 12}}{2 \cdot 1}$

Step 2.8.1.3

Subtract $12$ from $1$ .

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

Step 2.8.1.4

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

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

Step 2.8.1.5

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

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

Step 2.8.1.6

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

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

Step 2.8.2

Multiply $2$ by $1$ .

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

Step 2.8.3

Change the $\pm$ to $-$ .

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

Step 2.8.4

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

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

Step 2.8.5

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

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

Step 2.8.6

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

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

Step 2.8.7

Move the negative in front of the fraction.

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

Step 2.9

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{11}}{2}, - \frac{1 + i \sqrt{11}}{2}$

Step 3

Evaluate $y$ when $x = - \frac{1 - i \sqrt{11}}{2}$ .

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Step 3.1

Substitute $- \frac{1 - i \sqrt{11}}{2}$ for $x$ .

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

Step 3.2

Simplify ${(- \frac{1 - i \sqrt{11}}{2})}^{2} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$ .

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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 $- \frac{1 - i \sqrt{11}}{2}$ .

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

Step 3.2.1.1.2

Apply the product rule to $\frac{1 - i \sqrt{11}}{2}$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply $\frac{{(1 - i \sqrt{11})}^{2}}{2^{2}}$ by $1$ .

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

Step 3.2.1.4

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

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

Step 3.2.1.5

Rewrite ${(1 - i \sqrt{11})}^{2}$ as $(1 - i \sqrt{11}) (1 - i \sqrt{11})$ .

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

Step 3.2.1.6

Expand $(1 - i \sqrt{11}) (1 - i \sqrt{11})$ using the FOIL Method.

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Step 3.2.1.6.1

Apply the distributive property.

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

Step 3.2.1.6.2

Apply the distributive property.

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

Step 3.2.1.6.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 3.2.1.7.1.2

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

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

Step 3.2.1.7.1.3

Multiply $- 1$ by $1$ .

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

Step 3.2.1.7.1.4

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

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Step 3.2.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.7.1.4.2

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

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

Step 3.2.1.7.1.4.3

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

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

Step 3.2.1.7.1.4.4

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

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

Step 3.2.1.7.1.4.5

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

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

Step 3.2.1.7.1.4.6

Add $1$ and $1$ .

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

Step 3.2.1.7.1.4.7

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

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

Step 3.2.1.7.1.4.8

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

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

Step 3.2.1.7.1.4.9

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

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

Step 3.2.1.7.1.4.10

Add $1$ and $1$ .

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

Step 3.2.1.7.1.5

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

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Step 3.2.1.7.1.5.1

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

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

Step 3.2.1.7.1.5.2

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

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

Step 3.2.1.7.1.5.3

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

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

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.

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

Step 3.2.1.7.1.5.4.2

Rewrite the expression.

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

Step 3.2.1.7.1.5.5

Evaluate the exponent.

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

Step 3.2.1.7.1.6

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

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

Step 3.2.1.7.1.7

Multiply $11$ by $- 1$ .

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

Step 3.2.1.7.2

Subtract $11$ from $1$ .

$y = \frac{- i \sqrt{11} - i \sqrt{11} - 10}{4} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

Step 3.2.1.7.3

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

$y = \frac{- 2 i \sqrt{11} - 10}{4} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

Step 3.2.1.8

Reorder $- 2 i \sqrt{11}$ and $- 10$ .

$y = \frac{- 10 - 2 i \sqrt{11}}{4} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

Step 3.2.1.9

Cancel the common factor of $- 10 - 2 i \sqrt{11}$ and $4$ .

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Step 3.2.1.9.1

Factor $2$ out of $- 10$ .

$y = \frac{2 (- 5) - 2 i \sqrt{11}}{4} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

Step 3.2.1.9.2

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

$y = \frac{2 (- 5) + 2 (- i \sqrt{11})}{4} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

Step 3.2.1.9.3

Factor $2$ out of $2 (- 5) + 2 (- i \sqrt{11})$ .

$y = \frac{2 (- 5 - i \sqrt{11})}{4} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

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 $4$ .

$y = \frac{2 (- 5 - i \sqrt{11})}{2 \cdot 2} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

Step 3.2.1.9.4.2

Cancel the common factor.

$y = \frac{2 (- 5 - i \sqrt{11})}{2 \cdot 2} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

Step 3.2.1.9.4.3

Rewrite the expression.

$y = \frac{- 5 - i \sqrt{11}}{2} + 2 (- \frac{1 - i \sqrt{11}}{2}) - 4$

Step 3.2.1.10

Cancel the common factor of $2$ .

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Step 3.2.1.10.1

Move the leading negative in $- \frac{1 - i \sqrt{11}}{2}$ into the numerator.

$y = \frac{- 5 - i \sqrt{11}}{2} + 2 \frac{- (1 - i \sqrt{11})}{2} - 4$

Step 3.2.1.10.2

Cancel the common factor.

$y = \frac{- 5 - i \sqrt{11}}{2} + 2 \frac{- (1 - i \sqrt{11})}{2} - 4$

Step 3.2.1.10.3

Rewrite the expression.

$y = \frac{- 5 - i \sqrt{11}}{2} - (1 - i \sqrt{11}) - 4$

Step 3.2.1.11

Apply the distributive property.

$y = \frac{- 5 - i \sqrt{11}}{2} - 1 \cdot 1 - (- i \sqrt{11}) - 4$

Step 3.2.1.12

Multiply $- 1$ by $1$ .

$y = \frac{- 5 - i \sqrt{11}}{2} - 1 - (- i \sqrt{11}) - 4$

Step 3.2.1.13

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

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Step 3.2.1.13.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 5 - i \sqrt{11}}{2} - 1 + 1 (i \sqrt{11}) - 4$

Step 3.2.1.13.2

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

$y = \frac{- 5 - i \sqrt{11}}{2} - 1 + \sqrt{11} i - 4$

Step 3.2.2

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

$y = \frac{- 5 - i \sqrt{11}}{2} - 1 \cdot \frac{2}{2} + \sqrt{11} i - 4$

Step 3.2.3

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

$y = \frac{- 5 - i \sqrt{11}}{2} + \frac{- 1 \cdot 2}{2} + \sqrt{11} i - 4$

Step 3.2.4

Combine the numerators over the common denominator.

$y = \frac{- 7 - i \sqrt{11}}{2} + \sqrt{11} i - 4$

Step 3.2.5

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

$y = \frac{- 7 - i \sqrt{11}}{2} + \sqrt{11} i \cdot \frac{2}{2} - 4$

Step 3.2.6

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

$y = \frac{- 7 - i \sqrt{11}}{2} + \frac{\sqrt{11} i \cdot 2}{2} - 4$

Step 3.2.7

Combine the numerators over the common denominator.

$y = \frac{- 7 + i \sqrt{11}}{2} - 4$

Step 3.2.8

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 7 + i \sqrt{11}}{2} - 4 \cdot \frac{2}{2}$

Step 3.2.9

Combine $- 4$ and $\frac{2}{2}$ .

$y = \frac{- 7 + i \sqrt{11}}{2} + \frac{- 4 \cdot 2}{2}$

Step 3.2.10

Combine the numerators over the common denominator.

$y = \frac{- 15 + i \sqrt{11}}{2}$

Step 3.2.11

Rewrite $- 15$ as $- 1 (15)$ .

$y = \frac{- 1 (15) + i \sqrt{11}}{2}$

Step 3.2.12

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

$y = \frac{- 1 (15) - (- i \sqrt{11})}{2}$

Step 3.2.13

Factor $- 1$ out of $- 1 (15) - (- i \sqrt{11})$ .

$y = \frac{- 1 (15 - i \sqrt{11})}{2}$

Step 3.2.14

Move the negative in front of the fraction.

$y = - \frac{15 - i \sqrt{11}}{2}$

Step 4

Evaluate $y$ when $x = - \frac{1 + i \sqrt{11}}{2}$ .

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Step 4.1

Substitute $- \frac{1 + i \sqrt{11}}{2}$ for $x$ .

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

Step 4.2

Simplify ${(- \frac{1 + i \sqrt{11}}{2})}^{2} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$ .

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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 $- \frac{1 + i \sqrt{11}}{2}$ .

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

Step 4.2.1.1.2

Apply the product rule to $\frac{1 + i \sqrt{11}}{2}$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Multiply $\frac{{(1 + i \sqrt{11})}^{2}}{2^{2}}$ by $1$ .

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

Step 4.2.1.4

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

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

Step 4.2.1.5

Rewrite ${(1 + i \sqrt{11})}^{2}$ as $(1 + i \sqrt{11}) (1 + i \sqrt{11})$ .

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

Step 4.2.1.6

Expand $(1 + i \sqrt{11}) (1 + i \sqrt{11})$ using the FOIL Method.

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Step 4.2.1.6.1

Apply the distributive property.

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

Step 4.2.1.6.2

Apply the distributive property.

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

Step 4.2.1.6.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 4.2.1.7.1.2

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

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

Step 4.2.1.7.1.3

Multiply $i$ by $1$ .

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

Step 4.2.1.7.1.4

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

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Step 4.2.1.7.1.4.1

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

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

Step 4.2.1.7.1.4.2

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

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

Step 4.2.1.7.1.4.3

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

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

Step 4.2.1.7.1.4.4

Add $1$ and $1$ .

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

Step 4.2.1.7.1.4.5

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

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

Step 4.2.1.7.1.4.6

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

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

Step 4.2.1.7.1.4.7

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

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

Step 4.2.1.7.1.4.8

Add $1$ and $1$ .

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

Step 4.2.1.7.1.5

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

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

Step 4.2.1.7.1.6

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

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Step 4.2.1.7.1.6.1

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

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

Step 4.2.1.7.1.6.2

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

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

Step 4.2.1.7.1.6.3

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

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

Step 4.2.1.7.1.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.7.1.6.4.1

Cancel the common factor.

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

Step 4.2.1.7.1.6.4.2

Rewrite the expression.

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

Step 4.2.1.7.1.6.5

Evaluate the exponent.

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

Step 4.2.1.7.1.7

Multiply $- 1$ by $11$ .

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

Step 4.2.1.7.2

Subtract $11$ from $1$ .

$y = \frac{i \sqrt{11} + i \sqrt{11} - 10}{4} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

Step 4.2.1.7.3

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

$y = \frac{2 i \sqrt{11} - 10}{4} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

Step 4.2.1.8

Reorder $2 i \sqrt{11}$ and $- 10$ .

$y = \frac{- 10 + 2 i \sqrt{11}}{4} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

Step 4.2.1.9

Cancel the common factor of $- 10 + 2 i \sqrt{11}$ and $4$ .

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Step 4.2.1.9.1

Factor $2$ out of $- 10$ .

$y = \frac{2 \cdot - 5 + 2 i \sqrt{11}}{4} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

Step 4.2.1.9.2

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

$y = \frac{2 \cdot - 5 + 2 (i \sqrt{11})}{4} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

Step 4.2.1.9.3

Factor $2$ out of $2 \cdot - 5 + 2 (i \sqrt{11})$ .

$y = \frac{2 \cdot (- 5 + i \sqrt{11})}{4} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

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 $4$ .

$y = \frac{2 \cdot (- 5 + i \sqrt{11})}{2 (2)} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

Step 4.2.1.9.4.2

Cancel the common factor.

$y = \frac{2 \cdot (- 5 + i \sqrt{11})}{2 \cdot 2} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

Step 4.2.1.9.4.3

Rewrite the expression.

$y = \frac{- 5 + i \sqrt{11}}{2} + 2 (- \frac{1 + i \sqrt{11}}{2}) - 4$

Step 4.2.1.10

Cancel the common factor of $2$ .

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Step 4.2.1.10.1

Move the leading negative in $- \frac{1 + i \sqrt{11}}{2}$ into the numerator.

$y = \frac{- 5 + i \sqrt{11}}{2} + 2 \frac{- (1 + i \sqrt{11})}{2} - 4$

Step 4.2.1.10.2

Cancel the common factor.

$y = \frac{- 5 + i \sqrt{11}}{2} + 2 \frac{- (1 + i \sqrt{11})}{2} - 4$

Step 4.2.1.10.3

Rewrite the expression.

$y = \frac{- 5 + i \sqrt{11}}{2} - (1 + i \sqrt{11}) - 4$

Step 4.2.1.11

Apply the distributive property.

$y = \frac{- 5 + i \sqrt{11}}{2} - 1 \cdot 1 - (i \sqrt{11}) - 4$

Step 4.2.1.12

Multiply $- 1$ by $1$ .

$y = \frac{- 5 + i \sqrt{11}}{2} - 1 - i \sqrt{11} - 4$

Step 4.2.2

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

$y = \frac{- 5 + i \sqrt{11}}{2} - 1 \cdot \frac{2}{2} - i \sqrt{11} - 4$

Step 4.2.3

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

$y = \frac{- 5 + i \sqrt{11}}{2} + \frac{- 1 \cdot 2}{2} - i \sqrt{11} - 4$

Step 4.2.4

Combine the numerators over the common denominator.

$y = \frac{- 7 + i \sqrt{11}}{2} - i \sqrt{11} - 4$

Step 4.2.5

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

$y = \frac{- 7 + i \sqrt{11}}{2} - i \sqrt{11} \cdot \frac{2}{2} - 4$

Step 4.2.6

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

$y = \frac{- 7 + i \sqrt{11}}{2} + \frac{- i \sqrt{11} \cdot 2}{2} - 4$

Step 4.2.7

Combine the numerators over the common denominator.

$y = \frac{- 7 - i \sqrt{11}}{2} - 4$

Step 4.2.8

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- 7 - i \sqrt{11}}{2} - 4 \cdot \frac{2}{2}$

Step 4.2.9

Combine $- 4$ and $\frac{2}{2}$ .

$y = \frac{- 7 - i \sqrt{11}}{2} + \frac{- 4 \cdot 2}{2}$

Step 4.2.10

Combine the numerators over the common denominator.

$y = \frac{- 15 - i \sqrt{11}}{2}$

Step 4.2.11

Rewrite $- 15$ as $- 1 (15)$ .

$y = \frac{- 1 (15) - i \sqrt{11}}{2}$

Step 4.2.12

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

$y = \frac{- 1 (15) - (i \sqrt{11})}{2}$

Step 4.2.13

Factor $- 1$ out of $- 1 (15) - (i \sqrt{11})$ .

$y = \frac{- 1 (15 + i \sqrt{11})}{2}$

Step 4.2.14

Move the negative in front of the fraction.

$y = - \frac{15 + i \sqrt{11}}{2}$

Step 5

List all of the solutions.

$y = - \frac{15 - i \sqrt{11}}{2}, x = - \frac{1 - i \sqrt{11}}{2}$

$y = - \frac{15 + i \sqrt{11}}{2}, x = - \frac{1 + i \sqrt{11}}{2}$

Step 6