Algebra Examples

$1 + 2 i$ , $1 - 2 i$

Step 1

$x = 1 + 2 i$ and $x = 1 - 2 i$ are the two real distinct solutions for the quadratic equation, which means that $x - 1 + 2 i$ and $x - 1 - 2 i$ are the factors of the quadratic equation.

$(x - 1 + 2 i) (x - 1 - 2 i) = 0$

Step 2

Expand $(x - 1 + 2 i) (x - 1 - 2 i)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 1 + x (- 2 i) - 1 x - 1 \cdot - 1 - 1 (- 2 i) + 2 i x + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $x \cdot x + x \cdot - 1 + x (- 2 i) - 1 x - 1 \cdot - 1 - 1 (- 2 i) + 2 i x + 2 i \cdot - 1 + 2 i (- 2 i)$ .

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Step 3.1.1

Reorder the factors in the terms $x (- 2 i)$ and $2 i x$ .

$x \cdot x + x \cdot - 1 - 2 i x - 1 x - 1 \cdot - 1 - 1 (- 2 i) + 2 i x + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.1.2

Add $- 2 i x$ and $2 i x$ .

$x \cdot x + x \cdot - 1 + 0 - 1 x - 1 \cdot - 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.1.3

Add $x \cdot x + x \cdot - 1$ and $0$ .

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.2.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.2.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 1 x - 1 \cdot - 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.2.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - x - 1 \cdot - 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.2.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - 1 (- 2 i) + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.2.6

Multiply $- 2$ by $- 1$ .

$x^{2} - x - x + 1 + 2 i + 2 i \cdot - 1 + 2 i (- 2 i) = 0$

Step 3.2.7

Multiply $- 1$ by $2$ .

$x^{2} - x - x + 1 + 2 i - 2 i + 2 i (- 2 i) = 0$

Step 3.2.8

Multiply $2 i (- 2 i)$ .

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Step 3.2.8.1

Multiply $- 2$ by $2$ .

$x^{2} - x - x + 1 + 2 i - 2 i - 4 i i = 0$

Step 3.2.8.2

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

$x^{2} - x - x + 1 + 2 i - 2 i - 4 (i i) = 0$

Step 3.2.8.3

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

$x^{2} - x - x + 1 + 2 i - 2 i - 4 (i i) = 0$

Step 3.2.8.4

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

$x^{2} - x - x + 1 + 2 i - 2 i - 4 i^{1 + 1} = 0$

Step 3.2.8.5

Add $1$ and $1$ .

$x^{2} - x - x + 1 + 2 i - 2 i - 4 i^{2} = 0$

Step 3.2.9

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

$x^{2} - x - x + 1 + 2 i - 2 i - 4 \cdot - 1 = 0$

Step 3.2.10

Multiply $- 4$ by $- 1$ .

$x^{2} - x - x + 1 + 2 i - 2 i + 4 = 0$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Combine the opposite terms in $x^{2} - x - x + 1 + 2 i - 2 i + 4$ .

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Step 3.3.1.1

Subtract $2 i$ from $2 i$ .

$x^{2} - x - x + 1 + 0 + 4 = 0$

Step 3.3.1.2

Add $x^{2} - x - x + 1$ and $0$ .

$x^{2} - x - x + 1 + 4 = 0$

Step 3.3.2

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 + 4 = 0$

Step 3.3.3

Add $1$ and $4$ .

$x^{2} - 2 x + 5 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${1 + 2 i, 1 - 2 i}$ is $y = x^{2} - 2 x + 5$ .

$y = x^{2} - 2 x + 5$

Step 5