Algebra Examples

$(15 - 4 i) (6 - 3 i) = a + b i$

Step 1

Rewrite the equation as $a + b i = (15 - 4 i) (6 - 3 i)$ .

$a + b i = (15 - 4 i) (6 - 3 i)$

Step 2

Simplify $(15 - 4 i) (6 - 3 i)$ .

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Step 2.1

Rewrite.

$a + b i = 0 + 0 + (15 - 4 i) (6 - 3 i)$

Step 2.2

Simplify by adding zeros.

$a + b i = (15 - 4 i) (6 - 3 i)$

Step 2.3

Expand $(15 - 4 i) (6 - 3 i)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$a + b i = 15 (6 - 3 i) - 4 i (6 - 3 i)$

Step 2.3.2

Apply the distributive property.

$a + b i = 15 \cdot 6 + 15 (- 3 i) - 4 i (6 - 3 i)$

Step 2.3.3

Apply the distributive property.

$a + b i = 15 \cdot 6 + 15 (- 3 i) - 4 i \cdot 6 - 4 i (- 3 i)$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $15$ by $6$ .

$a + b i = 90 + 15 (- 3 i) - 4 i \cdot 6 - 4 i (- 3 i)$

Step 2.4.1.2

Multiply $- 3$ by $15$ .

$a + b i = 90 - 45 i - 4 i \cdot 6 - 4 i (- 3 i)$

Step 2.4.1.3

Multiply $6$ by $- 4$ .

$a + b i = 90 - 45 i - 24 i - 4 i (- 3 i)$

Step 2.4.1.4

Multiply $- 4 i (- 3 i)$ .

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Step 2.4.1.4.1

Multiply $- 3$ by $- 4$ .

$a + b i = 90 - 45 i - 24 i + 12 i i$

Step 2.4.1.4.2

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

$a + b i = 90 - 45 i - 24 i + 12 (i^{1} i)$

Step 2.4.1.4.3

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

$a + b i = 90 - 45 i - 24 i + 12 (i^{1} i^{1})$

Step 2.4.1.4.4

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

$a + b i = 90 - 45 i - 24 i + 12 i^{1 + 1}$

Step 2.4.1.4.5

Add $1$ and $1$ .

$a + b i = 90 - 45 i - 24 i + 12 i^{2}$

Step 2.4.1.5

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

$a + b i = 90 - 45 i - 24 i + 12 \cdot - 1$

Step 2.4.1.6

Multiply $12$ by $- 1$ .

$a + b i = 90 - 45 i - 24 i - 12$

Step 2.4.2

Subtract $12$ from $90$ .

$a + b i = 78 - 45 i - 24 i$

Step 2.4.3

Subtract $24 i$ from $- 45 i$ .

$a + b i = 78 - 69 i$

Step 3

Subtract $b i$ from both sides of the equation.

$a = 78 - 69 i - b i$