Trigonometry Examples

$(2 + i) x - i = 5 + i$

Step 1

Apply the distributive property.

$2 x + i x - i = 5 + i$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Add $i$ to both sides of the equation.

$2 x + i x = 5 + i + i$

Step 2.2

Add $i$ and $i$ .

$2 x + i x = 5 + 2 i$

Step 3

Factor $x$ out of $2 x + i x$ .

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Step 3.1

Factor $x$ out of $2 x$ .

$x \cdot 2 + i x = 5 + 2 i$

Step 3.2

Factor $x$ out of $i x$ .

$x \cdot 2 + x i = 5 + 2 i$

Step 3.3

Factor $x$ out of $x \cdot 2 + x i$ .

$x (2 + i) = 5 + 2 i$

Step 4

Divide each term in $x (2 + i) = 5 + 2 i$ by $2 + i$ and simplify.

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Step 4.1

Divide each term in $x (2 + i) = 5 + 2 i$ by $2 + i$ .

$\frac{x (2 + i)}{2 + i} = \frac{5}{2 + i} + \frac{2 i}{2 + i}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2 + i$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x (2 + i)}{2 + i} = \frac{5}{2 + i} + \frac{2 i}{2 + i}$

Step 4.2.1.2

Divide $x$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

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

Step 4.3.2

Multiply the numerator and denominator of $\frac{5 + 2 i}{2 + 1 i}$ by the conjugate of $2 + 1 i$ to make the denominator real.

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

Step 4.3.3

Multiply.

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Step 4.3.3.1

Combine.

$x = \frac{(5 + 2 i) (2 - i)}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2

Simplify the numerator.

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Step 4.3.3.2.1

Expand $(5 + 2 i) (2 - i)$ using the FOIL Method.

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Step 4.3.3.2.1.1

Apply the distributive property.

$x = \frac{5 (2 - i) + 2 i (2 - i)}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.1.2

Apply the distributive property.

$x = \frac{5 \cdot 2 + 5 (- i) + 2 i (2 - i)}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.1.3

Apply the distributive property.

$x = \frac{5 \cdot 2 + 5 (- i) + 2 i \cdot 2 + 2 i (- i)}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2

Simplify and combine like terms.

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Step 4.3.3.2.2.1

Simplify each term.

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Step 4.3.3.2.2.1.1

Multiply $5$ by $2$ .

$x = \frac{10 + 5 (- i) + 2 i \cdot 2 + 2 i (- i)}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.2

Multiply $- 1$ by $5$ .

$x = \frac{10 - 5 i + 2 i \cdot 2 + 2 i (- i)}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.3

Multiply $2$ by $2$ .

$x = \frac{10 - 5 i + 4 i + 2 i (- i)}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.4

Multiply $2 i (- i)$ .

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Step 4.3.3.2.2.1.4.1

Multiply $- 1$ by $2$ .

$x = \frac{10 - 5 i + 4 i - 2 i i}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.4.2

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

$x = \frac{10 - 5 i + 4 i - 2 (i^{1} i)}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.4.3

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

$x = \frac{10 - 5 i + 4 i - 2 (i^{1} i^{1})}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.4.4

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

$x = \frac{10 - 5 i + 4 i - 2 i^{1 + 1}}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.4.5

Add $1$ and $1$ .

$x = \frac{10 - 5 i + 4 i - 2 i^{2}}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.5

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

$x = \frac{10 - 5 i + 4 i - 2 \cdot - 1}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.1.6

Multiply $- 2$ by $- 1$ .

$x = \frac{10 - 5 i + 4 i + 2}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.2

Add $10$ and $2$ .

$x = \frac{12 - 5 i + 4 i}{(2 + 1 i) (2 - i)}$

Step 4.3.3.2.2.3

Add $- 5 i$ and $4 i$ .

$x = \frac{12 - i}{(2 + 1 i) (2 - i)}$

Step 4.3.3.3

Simplify the denominator.

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Step 4.3.3.3.1

Expand $(2 + 1 i) (2 - i)$ using the FOIL Method.

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Step 4.3.3.3.1.1

Apply the distributive property.

$x = \frac{12 - i}{2 (2 - i) + 1 i (2 - i)}$

Step 4.3.3.3.1.2

Apply the distributive property.

$x = \frac{12 - i}{2 \cdot 2 + 2 (- i) + 1 i (2 - i)}$

Step 4.3.3.3.1.3

Apply the distributive property.

$x = \frac{12 - i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 4.3.3.3.2

Simplify.

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Step 4.3.3.3.2.1

Multiply $2$ by $2$ .

$x = \frac{12 - i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 4.3.3.3.2.2

Multiply $- 1$ by $2$ .

$x = \frac{12 - i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)}$

Step 4.3.3.3.2.3

Multiply $2$ by $1$ .

$x = \frac{12 - i}{4 - 2 i + 2 i + 1 i (- i)}$

Step 4.3.3.3.2.4

Multiply $- 1$ by $1$ .

$x = \frac{12 - i}{4 - 2 i + 2 i - i i}$

Step 4.3.3.3.2.5

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

$x = \frac{12 - i}{4 - 2 i + 2 i - (i^{1} i)}$

Step 4.3.3.3.2.6

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

$x = \frac{12 - i}{4 - 2 i + 2 i - (i^{1} i^{1})}$

Step 4.3.3.3.2.7

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

$x = \frac{12 - i}{4 - 2 i + 2 i - i^{1 + 1}}$

Step 4.3.3.3.2.8

Add $1$ and $1$ .

$x = \frac{12 - i}{4 - 2 i + 2 i - i^{2}}$

Step 4.3.3.3.2.9

Add $- 2 i$ and $2 i$ .

$x = \frac{12 - i}{4 + 0 - i^{2}}$

Step 4.3.3.3.2.10

Add $4$ and $0$ .

$x = \frac{12 - i}{4 - i^{2}}$

Step 4.3.3.3.3

Simplify each term.

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Step 4.3.3.3.3.1

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

$x = \frac{12 - i}{4 - - 1}$

Step 4.3.3.3.3.2

Multiply $- 1$ by $- 1$ .

$x = \frac{12 - i}{4 + 1}$

Step 4.3.3.3.4

Add $4$ and $1$ .

$x = \frac{12 - i}{5}$

Step 4.3.4

Split the fraction $\frac{12 - i}{5}$ into two fractions.

$x = \frac{12}{5} + \frac{- i}{5}$

Step 4.3.5

Move the negative in front of the fraction.

$x = \frac{12}{5} - \frac{i}{5}$