Linear Algebra Examples

$u - 2 = \sqrt{- 2} u + 39$

Step 1

Simplify each term.

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Step 1.1

Rewrite $- 2$ as $- 1 (2)$ .

$u - 2 = \sqrt{- 1 (2)} u + 39$

Step 1.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$u - 2 = \sqrt{- 1} \cdot \sqrt{2} u + 39$

Step 1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$u - 2 = i \sqrt{2} u + 39$

Step 2

Subtract $i \sqrt{2} u$ from both sides of the equation.

$u - 2 - i \sqrt{2} u = 39$

Step 3

Move all terms not containing $u$ to the right side of the equation.

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Step 3.1

Add $2$ to both sides of the equation.

$u - i \sqrt{2} u = 39 + 2$

Step 3.2

Add $39$ and $2$ .

$u - i \sqrt{2} u = 41$

Step 4

Factor $u$ out of $u - i \sqrt{2} u$ .

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Step 4.1

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - i \sqrt{2} u = 41$

Step 4.2

Factor $u$ out of $- i \sqrt{2} u$ .

$u \cdot 1 + u (- i \sqrt{2}) = 41$

Step 4.3

Factor $u$ out of $u \cdot 1 + u (- i \sqrt{2})$ .

$u (1 - i \sqrt{2}) = 41$

Step 5

Divide each term in $u (1 - i \sqrt{2}) = 41$ by $1 - i \sqrt{2}$ and simplify.

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Step 5.1

Divide each term in $u (1 - i \sqrt{2}) = 41$ by $1 - i \sqrt{2}$ .

$\frac{u (1 - i \sqrt{2})}{1 - i \sqrt{2}} = \frac{41}{1 - i \sqrt{2}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $1 - i \sqrt{2}$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{u (1 - i \sqrt{2})}{1 - i \sqrt{2}} = \frac{41}{1 - i \sqrt{2}}$

Step 5.2.1.2

Divide $u$ by $1$ .

$u = \frac{41}{1 - i \sqrt{2}}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator and denominator of $\frac{41}{1 - 1.41421356 i}$ by the conjugate of $1 - 1.41421356 i$ to make the denominator real.

$u = \frac{41}{1 - 1.41421356 i} \cdot \frac{1 + 1.41421356 i}{1 + 1.41421356 i}$

Step 5.3.2

Multiply.

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Step 5.3.2.1

Combine.

$u = \frac{41 (1 + 1.41421356 i)}{(1 - 1.41421356 i) (1 + 1.41421356 i)}$

Step 5.3.2.2

Simplify the numerator.

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Step 5.3.2.2.1

Apply the distributive property.

$u = \frac{41 \cdot 1 + 41 (1.41421356 i)}{(1 - 1.41421356 i) (1 + 1.41421356 i)}$

Step 5.3.2.2.2

Multiply $41$ by $1$ .

$u = \frac{41 + 41 (1.41421356 i)}{(1 - 1.41421356 i) (1 + 1.41421356 i)}$

Step 5.3.2.2.3

Multiply $1.41421356$ by $41$ .

$u = \frac{41 + 57.98275605 i}{(1 - 1.41421356 i) (1 + 1.41421356 i)}$

Step 5.3.2.3

Simplify the denominator.

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Step 5.3.2.3.1

Expand $(1 - 1.41421356 i) (1 + 1.41421356 i)$ using the FOIL Method.

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Step 5.3.2.3.1.1

Apply the distributive property.

$u = \frac{41 + 57.98275605 i}{1 (1 + 1.41421356 i) - 1.41421356 i (1 + 1.41421356 i)}$

Step 5.3.2.3.1.2

Apply the distributive property.

$u = \frac{41 + 57.98275605 i}{1 \cdot 1 + 1 (1.41421356 i) - 1.41421356 i (1 + 1.41421356 i)}$

Step 5.3.2.3.1.3

Apply the distributive property.

$u = \frac{41 + 57.98275605 i}{1 \cdot 1 + 1 (1.41421356 i) - 1.41421356 i \cdot 1 - 1.41421356 i (1.41421356 i)}$

Step 5.3.2.3.2

Simplify.

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Step 5.3.2.3.2.1

Multiply $1$ by $1$ .

$u = \frac{41 + 57.98275605 i}{1 + 1 (1.41421356 i) - 1.41421356 i \cdot 1 - 1.41421356 i (1.41421356 i)}$

Step 5.3.2.3.2.2

Multiply $1.41421356$ by $1$ .

$u = \frac{41 + 57.98275605 i}{1 + 1.41421356 i - 1.41421356 i \cdot 1 - 1.41421356 i (1.41421356 i)}$

Step 5.3.2.3.2.3

Multiply $- 1.41421356$ by $1$ .

$u = \frac{41 + 57.98275605 i}{1 + 1.41421356 i - 1.41421356 i - 1.41421356 i (1.41421356 i)}$

Step 5.3.2.3.2.4

Multiply $1.41421356$ by $- 1.41421356$ .

$u = \frac{41 + 57.98275605 i}{1 + 1.41421356 i - 1.41421356 i - 2 i i}$

Step 5.3.2.3.2.5

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

$u = \frac{41 + 57.98275605 i}{1 + 1.41421356 i - 1.41421356 i - 2 (i^{1} i)}$

Step 5.3.2.3.2.6

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

$u = \frac{41 + 57.98275605 i}{1 + 1.41421356 i - 1.41421356 i - 2 (i^{1} i^{1})}$

Step 5.3.2.3.2.7

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

$u = \frac{41 + 57.98275605 i}{1 + 1.41421356 i - 1.41421356 i - 2 i^{1 + 1}}$

Step 5.3.2.3.2.8

Add $1$ and $1$ .

$u = \frac{41 + 57.98275605 i}{1 + 1.41421356 i - 1.41421356 i - 2 i^{2}}$

Step 5.3.2.3.2.9

Subtract $1.41421356 i$ from $1.41421356 i$ .

$u = \frac{41 + 57.98275605 i}{1 + 0 i - 2 i^{2}}$

Step 5.3.2.3.3

Simplify each term.

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Step 5.3.2.3.3.1

Multiply $0$ by $i$ .

$u = \frac{41 + 57.98275605 i}{1 + 0 - 2 i^{2}}$

Step 5.3.2.3.3.2

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

$u = \frac{41 + 57.98275605 i}{1 + 0 - 2 \cdot - 1}$

Step 5.3.2.3.3.3

Multiply $- 2$ by $- 1$ .

$u = \frac{41 + 57.98275605 i}{1 + 0 + 2}$

Step 5.3.2.3.4

Add $1$ and $0$ .

$u = \frac{41 + 57.98275605 i}{1 + 2}$

Step 5.3.2.3.5

Add $1$ and $2$ .

$u = \frac{41 + 57.98275605 i}{3}$

Step 5.3.3

Rewrite $41$ as $1 (41)$ .

$u = \frac{1 (41) + 57.98275605 i}{3}$

Step 5.3.4

Factor $1$ out of $57.98275605 i$ .

$u = \frac{1 (41) + 1 (57.98275605 i)}{3}$

Step 5.3.5

Factor $1$ out of $1 (41) + 1 (57.98275605 i)$ .

$u = \frac{1 (41 + 57.98275605 i)}{3}$

Step 5.3.6

Factor $3$ out of $3$ .

$u = \frac{1 (41 + 57.98275605 i)}{3 (1)}$

Step 5.3.7

Separate fractions.

$u = \frac{1}{3} \cdot \frac{41 + 57.98275605 i}{1}$

Step 5.3.8

Simplify the expression.

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Step 5.3.8.1

Divide $1$ by $3$ .

$u = 0.33333333 \frac{41 + 57.98275605 i}{1}$

Step 5.3.8.2

Divide $41 + 57.98275605 i$ by $1$ .

$u = 0.33333333 (41 + 57.98275605 i)$

Step 5.3.9

Apply the distributive property.

$u = 0.33333333 \cdot 41 + 0.33333333 (57.98275605 i)$

Step 5.3.10

Multiply.

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Step 5.3.10.1

Multiply $0.33333333$ by $41$ .

$u = 13.66666666 + 0.33333333 (57.98275605 i)$

Step 5.3.10.2

Multiply $57.98275605$ by $0.33333333$ .

$u = 13.66666666 + 19.32758535 i$

Step 6

The domain is the set of all valid $u$ values.

${u | u = 13.66666666 + 19.32758535 i}$

Step 7