Algebra Examples

${(x + (3 + 5 i))}^{2}$

Step 1

Simplify the expression.

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Step 1.1

Remove parentheses.

${(x + 3 + 5 i)}^{2}$

Step 1.2

Rewrite ${(x + 3 + 5 i)}^{2}$ as $(x + 3 + 5 i) (x + 3 + 5 i)$ .

$(x + 3 + 5 i) (x + 3 + 5 i)$

Step 2

Expand $(x + 3 + 5 i) (x + 3 + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot 3 + x (5 i) + 3 x + 3 \cdot 3 + 3 (5 i) + 5 i x + 5 i \cdot 3 + 5 i (5 i)$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 3 + x (5 i) + 3 x + 3 \cdot 3 + 3 (5 i) + 5 i x + 5 i \cdot 3 + 5 i (5 i)$

Step 3.1.2

Move $3$ to the left of $x$ .

$x^{2} + 3 \cdot x + x (5 i) + 3 x + 3 \cdot 3 + 3 (5 i) + 5 i x + 5 i \cdot 3 + 5 i (5 i)$

Step 3.1.3

Move $5$ to the left of $x$ .

$x^{2} + 3 x + 5 \cdot x i + 3 x + 3 \cdot 3 + 3 (5 i) + 5 i x + 5 i \cdot 3 + 5 i (5 i)$

Step 3.1.4

Multiply $3$ by $3$ .

$x^{2} + 3 x + 5 x i + 3 x + 9 + 3 (5 i) + 5 i x + 5 i \cdot 3 + 5 i (5 i)$

Step 3.1.5

Multiply $5$ by $3$ .

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 5 i \cdot 3 + 5 i (5 i)$

Step 3.1.6

Multiply $3$ by $5$ .

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 15 i + 5 i (5 i)$

Step 3.1.7

Multiply $5 i (5 i)$ .

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Step 3.1.7.1

Multiply $5$ by $5$ .

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 15 i + 25 i i$

Step 3.1.7.2

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

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 15 i + 25 (i^{1} i)$

Step 3.1.7.3

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

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 15 i + 25 (i^{1} i^{1})$

Step 3.1.7.4

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

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 15 i + 25 i^{1 + 1}$

Step 3.1.7.5

Add $1$ and $1$ .

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 15 i + 25 i^{2}$

Step 3.1.8

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

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 15 i + 25 \cdot - 1$

Step 3.1.9

Multiply $25$ by $- 1$ .

$x^{2} + 3 x + 5 x i + 3 x + 9 + 15 i + 5 i x + 15 i - 25$

Step 3.2

Add $3 x$ and $3 x$ .

$x^{2} + 5 x i + 6 x + 9 + 15 i + 5 i x + 15 i - 25$

Step 4

Add $5 x i$ and $5 i x$ .

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Step 4.1

Move $i$ .

$x^{2} + 5 x i + 5 x i + 6 x + 9 + 15 i + 15 i - 25$

Step 4.2

Add $5 x i$ and $5 x i$ .

$x^{2} + 10 x i + 6 x + 9 + 15 i + 15 i - 25$

Step 5

Subtract $25$ from $9$ .

$x^{2} + 10 x i + 6 x - 16 + 15 i + 15 i$

Step 6

Add $15 i$ and $15 i$ .

$x^{2} + 10 x i + 6 x - 16 + 30 i$