Precalculus Examples

${(4 + i)}^{2} - {(5 - i)}^{2}$

Step 1

Simplify each term.

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Step 1.1

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

$(4 + i) (4 + i) - {(5 - i)}^{2}$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$4 (4 + i) + i (4 + i) - {(5 - i)}^{2}$

Step 1.2.2

Apply the distributive property.

$4 \cdot 4 + 4 i + i (4 + i) - {(5 - i)}^{2}$

Step 1.2.3

Apply the distributive property.

$4 \cdot 4 + 4 i + i \cdot 4 + i i - {(5 - i)}^{2}$

$4 \cdot 4 + 4 i + i \cdot 4 + i i - {(5 - i)}^{2}$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $4$ by $4$ .

$16 + 4 i + i \cdot 4 + i i - {(5 - i)}^{2}$

Step 1.3.1.2

Move $4$ to the left of $i$ .

$16 + 4 i + 4 \cdot i + i i - {(5 - i)}^{2}$

Step 1.3.1.3

Multiply $i i$ .

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Step 1.3.1.3.1

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

$16 + 4 i + 4 i + i^{1} i - {(5 - i)}^{2}$

Step 1.3.1.3.2

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

$16 + 4 i + 4 i + i^{1} i^{1} - {(5 - i)}^{2}$

Step 1.3.1.3.3

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

$16 + 4 i + 4 i + i^{1 + 1} - {(5 - i)}^{2}$

Step 1.3.1.3.4

Add $1$ and $1$ .

$16 + 4 i + 4 i + i^{2} - {(5 - i)}^{2}$

$16 + 4 i + 4 i + i^{2} - {(5 - i)}^{2}$

Step 1.3.1.4

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

$16 + 4 i + 4 i - 1 - {(5 - i)}^{2}$

$16 + 4 i + 4 i - 1 - {(5 - i)}^{2}$

Step 1.3.2

Subtract $1$ from $16$ .

$15 + 4 i + 4 i - {(5 - i)}^{2}$

Step 1.3.3

Add $4 i$ and $4 i$ .

$15 + 8 i - {(5 - i)}^{2}$

$15 + 8 i - {(5 - i)}^{2}$

Step 1.4

Rewrite ${(5 - i)}^{2}$ as $(5 - i) (5 - i)$ .

$15 + 8 i - ((5 - i) (5 - i))$

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

$15 + 8 i - (5 (5 - i) - i (5 - i))$

Step 1.5.2

Apply the distributive property.

$15 + 8 i - (5 \cdot 5 + 5 (- i) - i (5 - i))$

Step 1.5.3

Apply the distributive property.

$15 + 8 i - (5 \cdot 5 + 5 (- i) - i \cdot 5 - i (- i))$

$15 + 8 i - (5 \cdot 5 + 5 (- i) - i \cdot 5 - i (- i))$

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $5$ by $5$ .

$15 + 8 i - (25 + 5 (- i) - i \cdot 5 - i (- i))$

Step 1.6.1.2

Multiply $- 1$ by $5$ .

$15 + 8 i - (25 - 5 i - i \cdot 5 - i (- i))$

Step 1.6.1.3

Multiply $5$ by $- 1$ .

$15 + 8 i - (25 - 5 i - 5 i - i (- i))$

Step 1.6.1.4

Multiply $- i (- i)$ .

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Step 1.6.1.4.1

Multiply $- 1$ by $- 1$ .

$15 + 8 i - (25 - 5 i - 5 i + 1 i i)$

Step 1.6.1.4.2

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

$15 + 8 i - (25 - 5 i - 5 i + 1 (i^{1} i))$

Step 1.6.1.4.3

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

$15 + 8 i - (25 - 5 i - 5 i + 1 (i^{1} i^{1}))$

Step 1.6.1.4.4

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

$15 + 8 i - (25 - 5 i - 5 i + 1 i^{1 + 1})$

Step 1.6.1.4.5

Add $1$ and $1$ .

$15 + 8 i - (25 - 5 i - 5 i + 1 i^{2})$

Step 1.6.1.4.6

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

$15 + 8 i - (25 - 5 i - 5 i + i^{2})$

$15 + 8 i - (25 - 5 i - 5 i + i^{2})$

Step 1.6.1.5

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

$15 + 8 i - (25 - 5 i - 5 i - 1)$

$15 + 8 i - (25 - 5 i - 5 i - 1)$

Step 1.6.2

Subtract $1$ from $25$ .

$15 + 8 i - (24 - 5 i - 5 i)$

Step 1.6.3

Subtract $5 i$ from $- 5 i$ .

$15 + 8 i - (24 - 10 i)$

$15 + 8 i - (24 - 10 i)$

Step 1.7

Apply the distributive property.

$15 + 8 i - 1 \cdot 24 - (- 10 i)$

Step 1.8

Multiply $- 1$ by $24$ .

$15 + 8 i - 24 - (- 10 i)$

Step 1.9

Multiply $- 10$ by $- 1$ .

$15 + 8 i - 24 + 10 i$

$15 + 8 i - 24 + 10 i$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $24$ from $15$ .

$- 9 + 8 i + 10 i$

Step 2.2

Add $8 i$ and $10 i$ .

$- 9 + 18 i$

$- 9 + 18 i$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$