Algebra Examples

$y = - x + 3$ $y = 2 x^{2} - 3 x + 4$

Step 1

Eliminate the equal sides of each equation and combine.

$- x + 3 = 2 x^{2} - 3 x + 4$

Step 2

Solve $- x + 3 = 2 x^{2} - 3 x + 4$ 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.

$2 x^{2} - 3 x + 4 = - x + 3$

Step 2.2

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

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Step 2.2.1

Add $x$ to both sides of the equation.

$2 x^{2} - 3 x + 4 + x = 3$

Step 2.2.2

Add $- 3 x$ and $x$ .

$2 x^{2} - 2 x + 4 = 3$

Step 2.3

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

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Step 2.3.1

Subtract $3$ from both sides of the equation.

$2 x^{2} - 2 x + 4 - 3 = 0$

Step 2.3.2

Subtract $3$ from $4$ .

$2 x^{2} - 2 x + 1 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 2 x^{2} - 3 x + 4$

Step 2.5

Substitute the values $a = 2$ , $b = - 2$ , and $c = 1$ into the quadratic formula and solve for $x$ .

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

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

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

Step 2.6.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 1}}{2 \cdot 2}$

Step 2.6.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 2}$

Step 2.6.1.3

Subtract $8$ from $4$ .

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

Step 2.6.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.6.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 2}$

Step 2.6.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 2}$

Step 2.6.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 2}$

Step 2.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 2}$

Step 2.6.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i}{4}$

Step 2.6.3

Simplify $\frac{2 \pm 2 i}{4}$ .

$x = \frac{1 \pm i}{2}$

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

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

Step 2.7.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 1}}{2 \cdot 2}$

Step 2.7.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 2}$

Step 2.7.1.3

Subtract $8$ from $4$ .

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

Step 2.7.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.7.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.7.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 2}$

Step 2.7.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 2}$

Step 2.7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 2}$

Step 2.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 2}$

Step 2.7.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i}{4}$

Step 2.7.3

Simplify $\frac{2 \pm 2 i}{4}$ .

$x = \frac{1 \pm i}{2}$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = \frac{1 + i}{2}$

Step 2.7.5

Split the fraction $\frac{1 + i}{2}$ into two fractions.

$x = \frac{1}{2} + \frac{i}{2}$

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

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

Step 2.8.1.2

Multiply $- 4 \cdot 2 \cdot 1$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 1}}{2 \cdot 2}$

Step 2.8.1.2.2

Multiply $- 8$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 2}$

Step 2.8.1.3

Subtract $8$ from $4$ .

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

Step 2.8.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.8.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.8.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 2}$

Step 2.8.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 2}$

Step 2.8.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 2}$

Step 2.8.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 2}$

Step 2.8.2

Multiply $2$ by $2$ .

$x = \frac{2 \pm 2 i}{4}$

Step 2.8.3

Simplify $\frac{2 \pm 2 i}{4}$ .

$x = \frac{1 \pm i}{2}$

Step 2.8.4

Change the $\pm$ to $-$ .

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

Step 2.8.5

Split the fraction $\frac{1 - i}{2}$ into two fractions.

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

Step 2.8.6

Move the negative in front of the fraction.

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

Step 2.9

The final answer is the combination of both solutions.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify $2 {(\frac{1}{2} + \frac{i}{2})}^{2} - 3 (\frac{1}{2} + \frac{i}{2}) + 4$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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

Step 3.2.1.2

Expand $(\frac{1}{2} + \frac{i}{2}) (\frac{1}{2} + \frac{i}{2})$ using the FOIL Method.

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Step 3.2.1.2.1

Apply the distributive property.

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

Step 3.2.1.2.2

Apply the distributive property.

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

Step 3.2.1.2.3

Apply the distributive property.

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

Step 3.2.1.3

Simplify and combine like terms.

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Step 3.2.1.3.1

Simplify each term.

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Step 3.2.1.3.1.1

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 3.2.1.3.1.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 3.2.1.3.1.1.2

Multiply $2$ by $2$ .

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

Step 3.2.1.3.1.2

Multiply $\frac{1}{2} \cdot \frac{i}{2}$ .

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Step 3.2.1.3.1.2.1

Multiply $\frac{1}{2}$ by $\frac{i}{2}$ .

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

Step 3.2.1.3.1.2.2

Multiply $2$ by $2$ .

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

Step 3.2.1.3.1.3

Multiply $\frac{i}{2} \cdot \frac{1}{2}$ .

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Step 3.2.1.3.1.3.1

Multiply $\frac{i}{2}$ by $\frac{1}{2}$ .

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

Step 3.2.1.3.1.3.2

Multiply $2$ by $2$ .

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

Step 3.2.1.3.1.4

Multiply $\frac{i}{2} \cdot \frac{i}{2}$ .

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Step 3.2.1.3.1.4.1

Multiply $\frac{i}{2}$ by $\frac{i}{2}$ .

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

Step 3.2.1.3.1.4.2

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

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

Step 3.2.1.3.1.4.3

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

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

Step 3.2.1.3.1.4.4

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

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

Step 3.2.1.3.1.4.5

Add $1$ and $1$ .

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

Step 3.2.1.3.1.4.6

Multiply $2$ by $2$ .

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

Step 3.2.1.3.1.5

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

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

Step 3.2.1.3.1.6

Move the negative in front of the fraction.

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

Step 3.2.1.3.2

Combine the numerators over the common denominator.

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

Step 3.2.1.3.3

Subtract $1$ from $1$ .

$y = 2 (\frac{0}{4} + \frac{i}{4} + \frac{i}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Step 3.2.1.3.4

Combine the numerators over the common denominator.

$y = 2 (\frac{0}{4} + \frac{2 i}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Step 3.2.1.4

Simplify each term.

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Step 3.2.1.4.1

Divide $0$ by $4$ .

$y = 2 (0 + \frac{2 i}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Step 3.2.1.4.2

Cancel the common factor of $2$ and $4$ .

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Step 3.2.1.4.2.1

Factor $2$ out of $2 i$ .

$y = 2 (0 + \frac{2 (i)}{4}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Step 3.2.1.4.2.2

Cancel the common factors.

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Step 3.2.1.4.2.2.1

Factor $2$ out of $4$ .

$y = 2 (0 + \frac{2 i}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Step 3.2.1.4.2.2.2

Cancel the common factor.

$y = 2 (0 + \frac{2 i}{2 \cdot 2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Step 3.2.1.4.2.2.3

Rewrite the expression.

$y = 2 (0 + \frac{i}{2}) - 3 (\frac{1}{2} + \frac{i}{2}) + 4$

Step 3.2.1.5

Add $0$ and $\frac{i}{2}$ .

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

Step 3.2.1.6

Cancel the common factor of $2$ .

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Step 3.2.1.6.1

Cancel the common factor.

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

Step 3.2.1.6.2

Rewrite the expression.

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

Step 3.2.1.7

Apply the distributive property.

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

Step 3.2.1.8

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 3.2.1.9

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

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

Step 3.2.1.10

Simplify each term.

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Step 3.2.1.10.1

Move the negative in front of the fraction.

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

Step 3.2.1.10.2

Move the negative in front of the fraction.

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Combine the numerators over the common denominator.

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

Step 3.2.4.2

Move the negative in front of the fraction.

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

Combine the numerators over the common denominator.

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

Step 3.2.8

Simplify the numerator.

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Step 3.2.8.1

Multiply $4$ by $2$ .

$y = - \frac{i}{2} + \frac{- 3 + 8}{2}$

Step 3.2.8.2

Add $- 3$ and $8$ .

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

Step 3.2.9

Reorder $- \frac{i}{2}$ and $\frac{5}{2}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify $2 {(\frac{1}{2} - \frac{i}{2})}^{2} - 3 (\frac{1}{2} - \frac{i}{2}) + 4$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

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

Step 4.2.1.2

Expand $(\frac{1}{2} - \frac{i}{2}) (\frac{1}{2} - \frac{i}{2})$ using the FOIL Method.

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Step 4.2.1.2.1

Apply the distributive property.

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

Step 4.2.1.2.2

Apply the distributive property.

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

Step 4.2.1.2.3

Apply the distributive property.

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

Step 4.2.1.3

Simplify and combine like terms.

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Step 4.2.1.3.1

Simplify each term.

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Step 4.2.1.3.1.1

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 4.2.1.3.1.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 4.2.1.3.1.1.2

Multiply $2$ by $2$ .

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

Step 4.2.1.3.1.2

Multiply $\frac{1}{2} (- \frac{i}{2})$ .

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Step 4.2.1.3.1.2.1

Multiply $\frac{1}{2}$ by $\frac{i}{2}$ .

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

Step 4.2.1.3.1.2.2

Multiply $2$ by $2$ .

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

Step 4.2.1.3.1.3

Multiply $- \frac{i}{2} \cdot \frac{1}{2}$ .

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Step 4.2.1.3.1.3.1

Multiply $\frac{1}{2}$ by $\frac{i}{2}$ .

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

Step 4.2.1.3.1.3.2

Multiply $2$ by $2$ .

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

Step 4.2.1.3.1.4

Multiply $- \frac{i}{2} (- \frac{i}{2})$ .

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Step 4.2.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.1.3.1.4.2

Multiply $\frac{i}{2}$ by $1$ .

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

Step 4.2.1.3.1.4.3

Multiply $\frac{i}{2}$ by $\frac{i}{2}$ .

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

Step 4.2.1.3.1.4.4

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

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

Step 4.2.1.3.1.4.5

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

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

Step 4.2.1.3.1.4.6

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

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

Step 4.2.1.3.1.4.7

Add $1$ and $1$ .

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

Step 4.2.1.3.1.4.8

Multiply $2$ by $2$ .

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

Step 4.2.1.3.1.5

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

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

Step 4.2.1.3.1.6

Move the negative in front of the fraction.

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

Step 4.2.1.3.2

Combine the numerators over the common denominator.

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

Step 4.2.1.3.3

Subtract $1$ from $1$ .

$y = 2 (\frac{0}{4} - \frac{i}{4} - \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Step 4.2.1.3.4

Subtract $\frac{i}{4}$ from $- \frac{i}{4}$ .

$y = 2 (\frac{0}{4} - 2 \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Step 4.2.1.4

Simplify each term.

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Step 4.2.1.4.1

Divide $0$ by $4$ .

$y = 2 (0 - 2 \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Factor $2$ out of $- 2$ .

$y = 2 (0 + 2 (- 1) \frac{i}{4}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Step 4.2.1.4.2.2

Factor $2$ out of $4$ .

$y = 2 (0 + 2 \cdot - 1 \frac{i}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Step 4.2.1.4.2.3

Cancel the common factor.

$y = 2 (0 + 2 \cdot - 1 \frac{i}{2 \cdot 2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Step 4.2.1.4.2.4

Rewrite the expression.

$y = 2 (0 - 1 \frac{i}{2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Step 4.2.1.4.3

Rewrite $- 1 \frac{i}{2}$ as $- \frac{i}{2}$ .

$y = 2 (0 - \frac{i}{2}) - 3 (\frac{1}{2} - \frac{i}{2}) + 4$

Step 4.2.1.5

Subtract $\frac{i}{2}$ from $0$ .

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

Step 4.2.1.6

Cancel the common factor of $2$ .

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Step 4.2.1.6.1

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

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

Step 4.2.1.6.2

Cancel the common factor.

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

Step 4.2.1.6.3

Rewrite the expression.

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

Step 4.2.1.7

Apply the distributive property.

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

Step 4.2.1.8

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 4.2.1.9

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

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Step 4.2.1.9.1

Multiply $- 1$ by $- 3$ .

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

Step 4.2.1.9.2

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

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

Step 4.2.1.10

Move the negative in front of the fraction.

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

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

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

Step 4.2.6

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

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

Step 4.2.7

Combine the numerators over the common denominator.

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

Step 4.2.8

Simplify the numerator.

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Step 4.2.8.1

Multiply $4$ by $2$ .

$y = \frac{i}{2} + \frac{- 3 + 8}{2}$

Step 4.2.8.2

Add $- 3$ and $8$ .

$y = \frac{i}{2} + \frac{5}{2}$

Step 4.2.9

Reorder $\frac{i}{2}$ and $\frac{5}{2}$ .

$y = \frac{5}{2} + \frac{i}{2}$

Step 5

List all of the solutions.

$y = \frac{5}{2} - \frac{i}{2}, x = \frac{1}{2} + \frac{i}{2}$

$y = \frac{5}{2} + \frac{i}{2}, x = \frac{1}{2} - \frac{i}{2}$

Step 6