Basic Math Examples

$z - 1 (\frac{1}{4} + 3 i) (z - (\frac{1}{4} - 3 i))$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$z + (- 1 (\frac{1}{4}) - 1 (3 i)) (z - (\frac{1}{4} - 3 i))$

Step 1.2

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

$z + (- (\frac{1}{4}) - 1 (3 i)) (z - (\frac{1}{4} - 3 i))$

Step 1.3

Multiply $3$ by $- 1$ .

$z + (- (\frac{1}{4}) - 3 i) (z - (\frac{1}{4} - 3 i))$

Step 1.4

Simplify each term.

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Step 1.4.1

Apply the distributive property.

$z + (- \frac{1}{4} - 3 i) (z - \frac{1}{4} - (- 3 i))$

Step 1.4.2

Multiply $- 3$ by $- 1$ .

$z + (- \frac{1}{4} - 3 i) (z - \frac{1}{4} + 3 i)$

Step 1.5

Expand $(- \frac{1}{4} - 3 i) (z - \frac{1}{4} + 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$z - \frac{1}{4} z - \frac{1}{4} (- \frac{1}{4}) - \frac{1}{4} (3 i) - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6

Simplify each term.

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Step 1.6.1

Combine $z$ and $\frac{1}{4}$ .

$z - \frac{z}{4} - \frac{1}{4} (- \frac{1}{4}) - \frac{1}{4} (3 i) - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.2

Multiply $- \frac{1}{4} (- \frac{1}{4})$ .

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Step 1.6.2.1

Multiply $- 1$ by $- 1$ .

$z - \frac{z}{4} + 1 (\frac{1}{4}) \frac{1}{4} - \frac{1}{4} (3 i) - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.2.2

Multiply $\frac{1}{4}$ by $1$ .

$z - \frac{z}{4} + \frac{1}{4} \cdot \frac{1}{4} - \frac{1}{4} (3 i) - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.2.3

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$z - \frac{z}{4} + \frac{1}{4 \cdot 4} - \frac{1}{4} (3 i) - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.2.4

Multiply $4$ by $4$ .

$z - \frac{z}{4} + \frac{1}{16} - \frac{1}{4} (3 i) - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.3

Multiply $- \frac{1}{4} (3 i)$ .

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Step 1.6.3.1

Multiply $3$ by $- 1$ .

$z - \frac{z}{4} + \frac{1}{16} - 3 (\frac{1}{4}) i - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.3.2

Combine $- 3$ and $\frac{1}{4}$ .

$z - \frac{z}{4} + \frac{1}{16} + \frac{- 3}{4} i - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.3.3

Combine $\frac{- 3}{4}$ and $i$ .

$z - \frac{z}{4} + \frac{1}{16} + \frac{- 3 i}{4} - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.4

Move the negative in front of the fraction.

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z - 3 i (- \frac{1}{4}) - 3 i (3 i)$

Step 1.6.5

Multiply $- 3 i (- \frac{1}{4})$ .

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Step 1.6.5.1

Multiply $- 1$ by $- 3$ .

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + 3 i \frac{1}{4} - 3 i (3 i)$

Step 1.6.5.2

Combine $\frac{1}{4}$ and $3$ .

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3}{4} i - 3 i (3 i)$

Step 1.6.5.3

Combine $\frac{3}{4}$ and $i$ .

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} - 3 i (3 i)$

Step 1.6.6

Multiply $- 3 i (3 i)$ .

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Step 1.6.6.1

Multiply $3$ by $- 3$ .

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} - 9 i i$

Step 1.6.6.2

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

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} - 9 (i^{1} i)$

Step 1.6.6.3

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

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} - 9 (i^{1} i^{1})$

Step 1.6.6.4

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

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} - 9 i^{1 + 1}$

Step 1.6.6.5

Add $1$ and $1$ .

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} - 9 i^{2}$

Step 1.6.7

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

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} - 9 \cdot - 1$

Step 1.6.8

Multiply $- 9$ by $- 1$ .

$z - \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} + 9$

Step 1.7

Combine the opposite terms in $- \frac{z}{4} + \frac{1}{16} - \frac{3 i}{4} - 3 i z + \frac{3 i}{4} + 9$ .

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Step 1.7.1

Add $- \frac{3 i}{4}$ and $\frac{3 i}{4}$ .

$z - \frac{z}{4} + \frac{1}{16} - 3 i z + 0 + 9$

Step 1.7.2

Add $- \frac{z}{4} + \frac{1}{16} - 3 i z$ and $0$ .

$z - \frac{z}{4} + \frac{1}{16} - 3 i z + 9$

Step 1.8

To write $9$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$z - \frac{z}{4} - 3 i z + \frac{1}{16} + 9 \cdot \frac{16}{16}$

Step 1.9

Combine $9$ and $\frac{16}{16}$ .

$z - \frac{z}{4} - 3 i z + \frac{1}{16} + \frac{9 \cdot 16}{16}$

Step 1.10

Combine the numerators over the common denominator.

$z - \frac{z}{4} - 3 i z + \frac{1 + 9 \cdot 16}{16}$

Step 1.11

Simplify the numerator.

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Step 1.11.1

Multiply $9$ by $16$ .

$z - \frac{z}{4} - 3 i z + \frac{1 + 144}{16}$

Step 1.11.2

Add $1$ and $144$ .

$z - \frac{z}{4} - 3 i z + \frac{145}{16}$

Step 2

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

$z \cdot \frac{4}{4} - \frac{z}{4} - 3 i z + \frac{145}{16}$

Step 3

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

$\frac{z \cdot 4}{4} - \frac{z}{4} - 3 i z + \frac{145}{16}$

Step 4

Combine the numerators over the common denominator.

$\frac{z \cdot 4 - z}{4} - 3 i z + \frac{145}{16}$

Step 5

Subtract $z$ from $z \cdot 4$ .

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Step 5.1

Reorder $z$ and $4$ .

$\frac{4 \cdot z - z}{4} - 3 i z + \frac{145}{16}$

Step 5.2

Subtract $z$ from $4 \cdot z$ .

$\frac{3 \cdot z}{4} - 3 i z + \frac{145}{16}$

Step 6

Move $i$ .

$\frac{3 z}{4} - 3 z i + \frac{145}{16}$