Precalculus Examples

${(4 - i)}^{2} - {(1 + 2 i)}^{2}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$(4 - i) (4 - i) - {(1 + 2 i)}^{2}$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$4 (4 - i) - i (4 - i) - {(1 + 2 i)}^{2}$

Step 1.2.2

Apply the distributive property.

$4 \cdot 4 + 4 (- i) - i (4 - i) - {(1 + 2 i)}^{2}$

Step 1.2.3

Apply the distributive property.

$4 \cdot 4 + 4 (- i) - i \cdot 4 - i (- i) - {(1 + 2 i)}^{2}$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $4$ by $4$ .

$16 + 4 (- i) - i \cdot 4 - i (- i) - {(1 + 2 i)}^{2}$

Step 1.3.1.2

Multiply $- 1$ by $4$ .

$16 - 4 i - i \cdot 4 - i (- i) - {(1 + 2 i)}^{2}$

Step 1.3.1.3

Multiply $4$ by $- 1$ .

$16 - 4 i - 4 i - i (- i) - {(1 + 2 i)}^{2}$

Step 1.3.1.4

Multiply $- i (- i)$ .

Tap for more steps...

Step 1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$16 - 4 i - 4 i + 1 i i - {(1 + 2 i)}^{2}$

Step 1.3.1.4.2

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

$16 - 4 i - 4 i + 1 (i^{1} i) - {(1 + 2 i)}^{2}$

Step 1.3.1.4.3

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

$16 - 4 i - 4 i + 1 (i^{1} i^{1}) - {(1 + 2 i)}^{2}$

Step 1.3.1.4.4

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

$16 - 4 i - 4 i + 1 i^{1 + 1} - {(1 + 2 i)}^{2}$

Step 1.3.1.4.5

Add $1$ and $1$ .

$16 - 4 i - 4 i + 1 i^{2} - {(1 + 2 i)}^{2}$

Step 1.3.1.4.6

Multiply $i^{2}$ by $1$ .

$16 - 4 i - 4 i + i^{2} - {(1 + 2 i)}^{2}$

Step 1.3.1.5

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

$16 - 4 i - 4 i - 1 - {(1 + 2 i)}^{2}$

Step 1.3.2

Subtract $1$ from $16$ .

$15 - 4 i - 4 i - {(1 + 2 i)}^{2}$

Step 1.3.3

Subtract $4 i$ from $- 4 i$ .

$15 - 8 i - {(1 + 2 i)}^{2}$

Step 1.4

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

$15 - 8 i - ((1 + 2 i) (1 + 2 i))$

Step 1.5

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

Tap for more steps...

Step 1.5.1

Apply the distributive property.

$15 - 8 i - (1 (1 + 2 i) + 2 i (1 + 2 i))$

Step 1.5.2

Apply the distributive property.

$15 - 8 i - (1 \cdot 1 + 1 (2 i) + 2 i (1 + 2 i))$

Step 1.5.3

Apply the distributive property.

$15 - 8 i - (1 \cdot 1 + 1 (2 i) + 2 i \cdot 1 + 2 i (2 i))$

Step 1.6

Simplify and combine like terms.

Tap for more steps...

Step 1.6.1

Simplify each term.

Tap for more steps...

Step 1.6.1.1

Multiply $1$ by $1$ .

$15 - 8 i - (1 + 1 (2 i) + 2 i \cdot 1 + 2 i (2 i))$

Step 1.6.1.2

Multiply $2 i$ by $1$ .

$15 - 8 i - (1 + 2 i + 2 i \cdot 1 + 2 i (2 i))$

Step 1.6.1.3

Multiply $2$ by $1$ .

$15 - 8 i - (1 + 2 i + 2 i + 2 i (2 i))$

Step 1.6.1.4

Multiply $2 i (2 i)$ .

Tap for more steps...

Step 1.6.1.4.1

Multiply $2$ by $2$ .

$15 - 8 i - (1 + 2 i + 2 i + 4 i i)$

Step 1.6.1.4.2

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

$15 - 8 i - (1 + 2 i + 2 i + 4 (i^{1} i))$

Step 1.6.1.4.3

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

$15 - 8 i - (1 + 2 i + 2 i + 4 (i^{1} i^{1}))$

Step 1.6.1.4.4

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

$15 - 8 i - (1 + 2 i + 2 i + 4 i^{1 + 1})$

Step 1.6.1.4.5

Add $1$ and $1$ .

$15 - 8 i - (1 + 2 i + 2 i + 4 i^{2})$

Step 1.6.1.5

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

$15 - 8 i - (1 + 2 i + 2 i + 4 \cdot - 1)$

Step 1.6.1.6

Multiply $4$ by $- 1$ .

$15 - 8 i - (1 + 2 i + 2 i - 4)$

Step 1.6.2

Subtract $4$ from $1$ .

$15 - 8 i - (- 3 + 2 i + 2 i)$

Step 1.6.3

Add $2 i$ and $2 i$ .

$15 - 8 i - (- 3 + 4 i)$

Step 1.7

Apply the distributive property.

$15 - 8 i - - 3 - (4 i)$

Step 1.8

Multiply $- 1$ by $- 3$ .

$15 - 8 i + 3 - (4 i)$

Step 1.9

Multiply $4$ by $- 1$ .

$15 - 8 i + 3 - 4 i$

Step 2

Simplify by adding terms.

Tap for more steps...

Step 2.1

Add $15$ and $3$ .

$18 - 8 i - 4 i$

Step 2.2

Subtract $4 i$ from $- 8 i$ .

$18 - 12 i$