Finite Math Examples

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

Step 1

Simplify ${(3 + 4 i)}^{2} (2 x - 2 y i)$ .

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Step 1.1

Rewrite.

$0 + 0 + {(3 + 4 i)}^{2} (2 x - 2 y i) = x + y i$

Step 1.2

Apply the distributive property.

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

Step 1.3

Move $2$ to the left of ${(3 + 4 i)}^{2}$ .

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

Step 1.4

Reorder factors in $2 {(3 + 4 i)}^{2} x + {(3 + 4 i)}^{2} (- 2 y i)$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(3 + 4 i)}^{2}$ as $(3 + 4 i) (3 + 4 i)$ .

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

Step 2.2.2

Expand $(3 + 4 i) (3 + 4 i)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

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

Step 2.2.2.2

Apply the distributive property.

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

Step 2.2.2.3

Apply the distributive property.

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

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $3$ by $3$ .

$2 x (9 + 3 (4 i) + 4 i \cdot 3 + 4 i (4 i)) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.2

Multiply $4$ by $3$ .

$2 x (9 + 12 i + 4 i \cdot 3 + 4 i (4 i)) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.3

Multiply $3$ by $4$ .

$2 x (9 + 12 i + 12 i + 4 i (4 i)) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.4

Multiply $4 i (4 i)$ .

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Step 2.2.3.1.4.1

Multiply $4$ by $4$ .

$2 x (9 + 12 i + 12 i + 16 i i) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.4.2

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

$2 x (9 + 12 i + 12 i + 16 (i^{1} i)) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.4.3

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

$2 x (9 + 12 i + 12 i + 16 (i^{1} i^{1})) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.4.4

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

$2 x (9 + 12 i + 12 i + 16 i^{1 + 1}) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.4.5

Add $1$ and $1$ .

$2 x (9 + 12 i + 12 i + 16 i^{2}) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.5

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

$2 x (9 + 12 i + 12 i + 16 \cdot - 1) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.1.6

Multiply $16$ by $- 1$ .

$2 x (9 + 12 i + 12 i - 16) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.2

Subtract $16$ from $9$ .

$2 x (- 7 + 12 i + 12 i) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.3.3

Add $12 i$ and $12 i$ .

$2 x (- 7 + 24 i) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.4

Apply the distributive property.

$2 x \cdot - 7 + 2 x (24 i) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.5

Multiply $- 7$ by $2$ .

$- 14 x + 2 x (24 i) - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.6

Multiply $24$ by $2$ .

$- 14 x + 48 x i - 2 y i {(3 + 4 i)}^{2} - y i = x$

Step 2.2.7

Rewrite ${(3 + 4 i)}^{2}$ as $(3 + 4 i) (3 + 4 i)$ .

$- 14 x + 48 x i - 2 y i ((3 + 4 i) (3 + 4 i)) - y i = x$

Step 2.2.8

Expand $(3 + 4 i) (3 + 4 i)$ using the FOIL Method.

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Step 2.2.8.1

Apply the distributive property.

$- 14 x + 48 x i - 2 y i (3 (3 + 4 i) + 4 i (3 + 4 i)) - y i = x$

Step 2.2.8.2

Apply the distributive property.

$- 14 x + 48 x i - 2 y i (3 \cdot 3 + 3 (4 i) + 4 i (3 + 4 i)) - y i = x$

Step 2.2.8.3

Apply the distributive property.

$- 14 x + 48 x i - 2 y i (3 \cdot 3 + 3 (4 i) + 4 i \cdot 3 + 4 i (4 i)) - y i = x$

Step 2.2.9

Simplify and combine like terms.

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Step 2.2.9.1

Simplify each term.

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Step 2.2.9.1.1

Multiply $3$ by $3$ .

$- 14 x + 48 x i - 2 y i (9 + 3 (4 i) + 4 i \cdot 3 + 4 i (4 i)) - y i = x$

Step 2.2.9.1.2

Multiply $4$ by $3$ .

$- 14 x + 48 x i - 2 y i (9 + 12 i + 4 i \cdot 3 + 4 i (4 i)) - y i = x$

Step 2.2.9.1.3

Multiply $3$ by $4$ .

$- 14 x + 48 x i - 2 y i (9 + 12 i + 12 i + 4 i (4 i)) - y i = x$

Step 2.2.9.1.4

Multiply $4 i (4 i)$ .

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Step 2.2.9.1.4.1

Multiply $4$ by $4$ .

$- 14 x + 48 x i - 2 y i (9 + 12 i + 12 i + 16 i i) - y i = x$

Step 2.2.9.1.4.2

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

$- 14 x + 48 x i - 2 y i (9 + 12 i + 12 i + 16 (i^{1} i)) - y i = x$

Step 2.2.9.1.4.3

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

$- 14 x + 48 x i - 2 y i (9 + 12 i + 12 i + 16 (i^{1} i^{1})) - y i = x$

Step 2.2.9.1.4.4

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

$- 14 x + 48 x i - 2 y i (9 + 12 i + 12 i + 16 i^{1 + 1}) - y i = x$

Step 2.2.9.1.4.5

Add $1$ and $1$ .

$- 14 x + 48 x i - 2 y i (9 + 12 i + 12 i + 16 i^{2}) - y i = x$

Step 2.2.9.1.5

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

$- 14 x + 48 x i - 2 y i (9 + 12 i + 12 i + 16 \cdot - 1) - y i = x$

Step 2.2.9.1.6

Multiply $16$ by $- 1$ .

$- 14 x + 48 x i - 2 y i (9 + 12 i + 12 i - 16) - y i = x$

Step 2.2.9.2

Subtract $16$ from $9$ .

$- 14 x + 48 x i - 2 y i (- 7 + 12 i + 12 i) - y i = x$

Step 2.2.9.3

Add $12 i$ and $12 i$ .

$- 14 x + 48 x i - 2 y i (- 7 + 24 i) - y i = x$

Step 2.2.10

Apply the distributive property.

$- 14 x + 48 x i - 2 y i \cdot - 7 - 2 y i (24 i) - y i = x$

Step 2.2.11

Multiply $- 7$ by $- 2$ .

$- 14 x + 48 x i + 14 y i - 2 y i (24 i) - y i = x$

Step 2.2.12

Multiply $- 2 y i (24 i)$ .

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Step 2.2.12.1

Multiply $24$ by $- 2$ .

$- 14 x + 48 x i + 14 y i - 48 y i i - y i = x$

Step 2.2.12.2

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

$- 14 x + 48 x i + 14 y i - 48 y (i^{1} i) - y i = x$

Step 2.2.12.3

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

$- 14 x + 48 x i + 14 y i - 48 y (i^{1} i^{1}) - y i = x$

Step 2.2.12.4

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

$- 14 x + 48 x i + 14 y i - 48 y i^{1 + 1} - y i = x$

Step 2.2.12.5

Add $1$ and $1$ .

$- 14 x + 48 x i + 14 y i - 48 y i^{2} - y i = x$

Step 2.2.13

Simplify each term.

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Step 2.2.13.1

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

$- 14 x + 48 x i + 14 y i - 48 y \cdot - 1 - y i = x$

Step 2.2.13.2

Multiply $- 1$ by $- 48$ .

$- 14 x + 48 x i + 14 y i + 48 y - y i = x$

Step 2.3

Subtract $y i$ from $14 y i$ .

$- 14 x + 48 x i + 13 y i + 48 y = x$

Step 3

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

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Step 3.1

Add $14 x$ to both sides of the equation.

$48 x i + 13 y i + 48 y = x + 14 x$

Step 3.2

Subtract $48 x i$ from both sides of the equation.

$13 y i + 48 y = x + 14 x - 48 x i$

Step 3.3

Add $x$ and $14 x$ .

$13 y i + 48 y = 15 x - 48 x i$

Step 4

Factor $y$ out of $13 y i + 48 y$ .

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Step 4.1

Factor $y$ out of $13 y i$ .

$y (13 i) + 48 y = 15 x - 48 x i$

Step 4.2

Factor $y$ out of $48 y$ .

$y (13 i) + y \cdot 48 = 15 x - 48 x i$

Step 4.3

Factor $y$ out of $y (13 i) + y \cdot 48$ .

$y (13 i + 48) = 15 x - 48 x i$

Step 5

Divide each term in $y (13 i + 48) = 15 x - 48 x i$ by $13 i + 48$ and simplify.

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Step 5.1

Divide each term in $y (13 i + 48) = 15 x - 48 x i$ by $13 i + 48$ .

$\frac{y (13 i + 48)}{13 i + 48} = \frac{15 x}{13 i + 48} + \frac{- 48 x i}{13 i + 48}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $13 i + 48$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{y (13 i + 48)}{13 i + 48} = \frac{15 x}{13 i + 48} + \frac{- 48 x i}{13 i + 48}$

Step 5.2.1.2

Divide $y$ by $1$ .

$y = \frac{15 x}{13 i + 48} + \frac{- 48 x i}{13 i + 48}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Combine the numerators over the common denominator.

$y = \frac{15 x - 48 x i}{13 i + 48}$

Step 5.3.2

Multiply the numerator and denominator of $\frac{15 x - 48 x i}{48 + 13 i}$ by the conjugate of $48 + 13 i$ to make the denominator real.

$y = \frac{15 x - 48 x i}{48 + 13 i} \cdot \frac{48 - 13 i}{48 - 13 i}$

Step 5.3.3

Multiply.

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Step 5.3.3.1

Combine.

$y = \frac{(15 x - 48 x i) (48 - 13 i)}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Expand $(15 x - 48 x i) (48 - 13 i)$ using the FOIL Method.

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Step 5.3.3.2.1.1

Apply the distributive property.

$y = \frac{15 x (48 - 13 i) - 48 x i (48 - 13 i)}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.1.2

Apply the distributive property.

$y = \frac{15 x \cdot 48 + 15 x (- 13 i) - 48 x i (48 - 13 i)}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.1.3

Apply the distributive property.

$y = \frac{15 x \cdot 48 + 15 x (- 13 i) - 48 x i \cdot 48 - 48 x i (- 13 i)}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2

Simplify and combine like terms.

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Step 5.3.3.2.2.1

Simplify each term.

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Step 5.3.3.2.2.1.1

Multiply $48$ by $15$ .

$y = \frac{720 x + 15 x (- 13 i) - 48 x i \cdot 48 - 48 x i (- 13 i)}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.2

Multiply $- 13$ by $15$ .

$y = \frac{720 x - 195 x i - 48 x i \cdot 48 - 48 x i (- 13 i)}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.3

Multiply $48$ by $- 48$ .

$y = \frac{720 x - 195 x i - 2304 x i - 48 x i (- 13 i)}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.4

Multiply $- 48 x i (- 13 i)$ .

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Step 5.3.3.2.2.1.4.1

Multiply $- 13$ by $- 48$ .

$y = \frac{720 x - 195 x i - 2304 x i + 624 x i i}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.4.2

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

$y = \frac{720 x - 195 x i - 2304 x i + 624 x (i^{1} i)}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.4.3

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

$y = \frac{720 x - 195 x i - 2304 x i + 624 x (i^{1} i^{1})}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.4.4

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

$y = \frac{720 x - 195 x i - 2304 x i + 624 x i^{1 + 1}}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.4.5

Add $1$ and $1$ .

$y = \frac{720 x - 195 x i - 2304 x i + 624 x i^{2}}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.5

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

$y = \frac{720 x - 195 x i - 2304 x i + 624 x \cdot - 1}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.1.6

Multiply $- 1$ by $624$ .

$y = \frac{720 x - 195 x i - 2304 x i - 624 x}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.2

Subtract $624 x$ from $720 x$ .

$y = \frac{- 195 x i - 2304 x i + 96 x}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.2.2.3

Subtract $2304 x i$ from $- 195 x i$ .

$y = \frac{- 2499 x i + 96 x}{(48 + 13 i) (48 - 13 i)}$

Step 5.3.3.3

Simplify the denominator.

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Step 5.3.3.3.1

Expand $(48 + 13 i) (48 - 13 i)$ using the FOIL Method.

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Step 5.3.3.3.1.1

Apply the distributive property.

$y = \frac{- 2499 x i + 96 x}{48 (48 - 13 i) + 13 i (48 - 13 i)}$

Step 5.3.3.3.1.2

Apply the distributive property.

$y = \frac{- 2499 x i + 96 x}{48 \cdot 48 + 48 (- 13 i) + 13 i (48 - 13 i)}$

Step 5.3.3.3.1.3

Apply the distributive property.

$y = \frac{- 2499 x i + 96 x}{48 \cdot 48 + 48 (- 13 i) + 13 i \cdot 48 + 13 i (- 13 i)}$

Step 5.3.3.3.2

Simplify.

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Step 5.3.3.3.2.1

Multiply $48$ by $48$ .

$y = \frac{- 2499 x i + 96 x}{2304 + 48 (- 13 i) + 13 i \cdot 48 + 13 i (- 13 i)}$

Step 5.3.3.3.2.2

Multiply $- 13$ by $48$ .

$y = \frac{- 2499 x i + 96 x}{2304 - 624 i + 13 i \cdot 48 + 13 i (- 13 i)}$

Step 5.3.3.3.2.3

Multiply $48$ by $13$ .

$y = \frac{- 2499 x i + 96 x}{2304 - 624 i + 624 i + 13 i (- 13 i)}$

Step 5.3.3.3.2.4

Multiply $- 13$ by $13$ .

$y = \frac{- 2499 x i + 96 x}{2304 - 624 i + 624 i - 169 i i}$

Step 5.3.3.3.2.5

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

$y = \frac{- 2499 x i + 96 x}{2304 - 624 i + 624 i - 169 (i^{1} i)}$

Step 5.3.3.3.2.6

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

$y = \frac{- 2499 x i + 96 x}{2304 - 624 i + 624 i - 169 (i^{1} i^{1})}$

Step 5.3.3.3.2.7

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

$y = \frac{- 2499 x i + 96 x}{2304 - 624 i + 624 i - 169 i^{1 + 1}}$

Step 5.3.3.3.2.8

Add $1$ and $1$ .

$y = \frac{- 2499 x i + 96 x}{2304 - 624 i + 624 i - 169 i^{2}}$

Step 5.3.3.3.2.9

Add $- 624 i$ and $624 i$ .

$y = \frac{- 2499 x i + 96 x}{2304 + 0 - 169 i^{2}}$

Step 5.3.3.3.2.10

Add $2304$ and $0$ .

$y = \frac{- 2499 x i + 96 x}{2304 - 169 i^{2}}$

Step 5.3.3.3.3

Simplify each term.

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Step 5.3.3.3.3.1

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

$y = \frac{- 2499 x i + 96 x}{2304 - 169 \cdot - 1}$

Step 5.3.3.3.3.2

Multiply $- 169$ by $- 1$ .

$y = \frac{- 2499 x i + 96 x}{2304 + 169}$

Step 5.3.3.3.4

Add $2304$ and $169$ .

$y = \frac{- 2499 x i + 96 x}{2473}$

Step 5.3.4

Factor $3 x$ out of $- 2499 x i + 96 x$ .

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Step 5.3.4.1

Factor $3 x$ out of $- 2499 x i$ .

$y = \frac{3 x (- 833 i) + 96 x}{2473}$

Step 5.3.4.2

Factor $3 x$ out of $96 x$ .

$y = \frac{3 x (- 833 i) + 3 x (32)}{2473}$

Step 5.3.4.3

Factor $3 x$ out of $3 x (- 833 i) + 3 x (32)$ .

$y = \frac{3 x (- 833 i + 32)}{2473}$

Step 5.3.5

Factor $- 1$ out of $- 833 i$ .

$y = \frac{3 x (- (833 i) + 32)}{2473}$

Step 5.3.6

Rewrite $32$ as $- 1 (- 32)$ .

$y = \frac{3 x (- (833 i) - 1 (- 32))}{2473}$

Step 5.3.7

Factor $- 1$ out of $- (833 i) - 1 (- 32)$ .

$y = \frac{3 x (- (833 i - 32))}{2473}$

Step 5.3.8

Simplify the expression.

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Step 5.3.8.1

Rewrite $- (833 i - 32)$ as $- 1 (833 i - 32)$ .

$y = \frac{3 x (- 1 (833 i - 32))}{2473}$

Step 5.3.8.2

Move the negative in front of the fraction.

$y = - \frac{(3 x) (833 i - 32)}{2473}$

Step 5.3.8.3

Reorder factors in $- \frac{(3 x) (833 i - 32)}{2473}$ .

$y = - \frac{3 x (833 i - 32)}{2473}$

Step 5.3.8.4

Reorder $833 i$ and $- 32$ .

$y = - \frac{3 x (- 32 + 833 i)}{2473}$