Algebra Examples

$1 + 4 i$ , $- 2 + 4 i$ , $- 2 - 3 i$ , $1 - 3 i$

Step 1

$x = 1 + 4 i$ and $x = - 2 + 4 i$ are the two real distinct solutions for the quadratic equation, which means that $x - 1 + 4 i$ and $x - - 2 + 4 i$ are the factors of the quadratic equation.

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

Step 2

Expand $(x - 1 + 4 i) (x + 2 + 4 i)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot 2 + x (4 i) - 1 x - 1 \cdot 2 - 1 (4 i) + 4 i x + 4 i \cdot 2 + 4 i (4 i) = 0$

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 $x$ .

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

Step 3.1.2

Move $2$ to the left of $x$ .

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Multiply $- 1$ by $2$ .

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

Step 3.1.6

Multiply $4$ by $- 1$ .

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

Step 3.1.7

Multiply $2$ by $4$ .

$x^{2} + 2 x + 4 x i - x - 2 - 4 i + 4 i x + 8 i + 4 i (4 i) = 0$

Step 3.1.8

Multiply $4 i (4 i)$ .

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Step 3.1.8.1

Multiply $4$ by $4$ .

$x^{2} + 2 x + 4 x i - x - 2 - 4 i + 4 i x + 8 i + 16 i i = 0$

Step 3.1.8.2

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

$x^{2} + 2 x + 4 x i - x - 2 - 4 i + 4 i x + 8 i + 16 (i i) = 0$

Step 3.1.8.3

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

$x^{2} + 2 x + 4 x i - x - 2 - 4 i + 4 i x + 8 i + 16 (i i) = 0$

Step 3.1.8.4

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

$x^{2} + 2 x + 4 x i - x - 2 - 4 i + 4 i x + 8 i + 16 i^{1 + 1} = 0$

Step 3.1.8.5

Add $1$ and $1$ .

$x^{2} + 2 x + 4 x i - x - 2 - 4 i + 4 i x + 8 i + 16 i^{2} = 0$

Step 3.1.9

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

$x^{2} + 2 x + 4 x i - x - 2 - 4 i + 4 i x + 8 i + 16 \cdot - 1 = 0$

Step 3.1.10

Multiply $16$ by $- 1$ .

$x^{2} + 2 x + 4 x i - x - 2 - 4 i + 4 i x + 8 i - 16 = 0$

Step 3.2

Subtract $x$ from $2 x$ .

$x^{2} + 4 x i + x - 2 - 4 i + 4 i x + 8 i - 16 = 0$

Step 4

Add $4 x i$ and $4 i x$ .

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Step 4.1

Move $i$ .

$x^{2} + 4 x i + 4 x i + x - 2 - 4 i + 8 i - 16 = 0$

Step 4.2

Add $4 x i$ and $4 x i$ .

$x^{2} + 8 x i + x - 2 - 4 i + 8 i - 16 = 0$

Step 5

Subtract $16$ from $- 2$ .

$x^{2} + 8 x i + x - 18 - 4 i + 8 i = 0$

Step 6

Add $- 4 i$ and $8 i$ .

$x^{2} + 8 x i + x - 18 + 4 i = 0$

Step 7

The standard quadratic equation using the given set of solutions ${1 + 4 i, - 2 + 4 i, - 2 - 3 i, 1 - 3 i}$ is $y = x^{2} + 8 x i + x - 18 + 4 i$ .

$y = x^{2} + 8 x i + x - 18 + 4 i$

Step 8