Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Expand $(5 + 2 i) (5 + 2 i)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $5$ by $5$ .

$25 + 5 (2 i) + 2 i \cdot 5 + 2 i (2 i) + {(2 - i)}^{2}$

Step 1.3.1.2

Multiply $2$ by $5$ .

$25 + 10 i + 2 i \cdot 5 + 2 i (2 i) + {(2 - i)}^{2}$

Step 1.3.1.3

Multiply $5$ by $2$ .

$25 + 10 i + 10 i + 2 i (2 i) + {(2 - i)}^{2}$

Step 1.3.1.4

Multiply $2 i (2 i)$ .

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Step 1.3.1.4.1

Multiply $2$ by $2$ .

$25 + 10 i + 10 i + 4 i i + {(2 - i)}^{2}$

Step 1.3.1.4.2

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

$25 + 10 i + 10 i + 4 (i^{1} i) + {(2 - i)}^{2}$

Step 1.3.1.4.3

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

$25 + 10 i + 10 i + 4 (i^{1} i^{1}) + {(2 - i)}^{2}$

Step 1.3.1.4.4

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

$25 + 10 i + 10 i + 4 i^{1 + 1} + {(2 - i)}^{2}$

Step 1.3.1.4.5

Add $1$ and $1$ .

$25 + 10 i + 10 i + 4 i^{2} + {(2 - i)}^{2}$

Step 1.3.1.5

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

$25 + 10 i + 10 i + 4 \cdot - 1 + {(2 - i)}^{2}$

Step 1.3.1.6

Multiply $4$ by $- 1$ .

$25 + 10 i + 10 i - 4 + {(2 - i)}^{2}$

Step 1.3.2

Subtract $4$ from $25$ .

$21 + 10 i + 10 i + {(2 - i)}^{2}$

Step 1.3.3

Add $10 i$ and $10 i$ .

$21 + 20 i + {(2 - i)}^{2}$

Step 1.4

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

$21 + 20 i + (2 - i) (2 - i)$

Step 1.5

Expand $(2 - i) (2 - i)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

$21 + 20 i + 2 (2 - i) - i (2 - i)$

Step 1.5.2

Apply the distributive property.

$21 + 20 i + 2 \cdot 2 + 2 (- i) - i (2 - i)$

Step 1.5.3

Apply the distributive property.

$21 + 20 i + 2 \cdot 2 + 2 (- i) - i \cdot 2 - i (- i)$

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 $2$ by $2$ .

$21 + 20 i + 4 + 2 (- i) - i \cdot 2 - i (- i)$

Step 1.6.1.2

Multiply $- 1$ by $2$ .

$21 + 20 i + 4 - 2 i - i \cdot 2 - i (- i)$

Step 1.6.1.3

Multiply $2$ by $- 1$ .

$21 + 20 i + 4 - 2 i - 2 i - i (- i)$

Step 1.6.1.4

Multiply $- i (- i)$ .

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Step 1.6.1.4.1

Multiply $- 1$ by $- 1$ .

$21 + 20 i + 4 - 2 i - 2 i + 1 i i$

Step 1.6.1.4.2

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

$21 + 20 i + 4 - 2 i - 2 i + 1 (i^{1} i)$

Step 1.6.1.4.3

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

$21 + 20 i + 4 - 2 i - 2 i + 1 (i^{1} i^{1})$

Step 1.6.1.4.4

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

$21 + 20 i + 4 - 2 i - 2 i + 1 i^{1 + 1}$

Step 1.6.1.4.5

Add $1$ and $1$ .

$21 + 20 i + 4 - 2 i - 2 i + 1 i^{2}$

Step 1.6.1.4.6

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

$21 + 20 i + 4 - 2 i - 2 i + i^{2}$

Step 1.6.1.5

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

$21 + 20 i + 4 - 2 i - 2 i - 1$

Step 1.6.2

Subtract $1$ from $4$ .

$21 + 20 i + 3 - 2 i - 2 i$

Step 1.6.3

Subtract $2 i$ from $- 2 i$ .

$21 + 20 i + 3 - 4 i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $21$ and $3$ .

$24 + 20 i - 4 i$

Step 2.2

Subtract $4 i$ from $20 i$ .

$24 + 16 i$