Precalculus Examples

$- 2 i$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = - 2 i$ was found by solving for $x$ when $x - (- 2 i) = y$ and $y = 0$ .

The factor is $x + 2 i$

Step 3

The root at $x = 2 i$ was found by solving for $x$ when $x - (2 i) = y$ and $y = 0$ .

The factor is $x - 2 i$

Step 4

Combine all the factors into a single equation.

$y = (x + 2 i) (x - 2 i)$

Step 5

Multiply all the factors to simplify the equation $y = (x + 2 i) (x - 2 i)$ .

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

$y = x (x - 2 i) + 2 i (x - 2 i)$

Step 5.1.2

Apply the distributive property.

$y = x \cdot x + x (- 2 i) + 2 i (x - 2 i)$

Step 5.1.3

Apply the distributive property.

$y = x \cdot x + x (- 2 i) + 2 i x + 2 i (- 2 i)$

Step 5.2

Simplify terms.

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Step 5.2.1

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

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Step 5.2.1.1

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

$y = x \cdot x - 2 i x + 2 i x + 2 i (- 2 i)$

Step 5.2.1.2

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

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

Step 5.2.1.3

Add $x \cdot x$ and $0$ .

$y = x \cdot x + 2 i (- 2 i)$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply $x$ by $x$ .

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

Step 5.2.2.2

Multiply $2 i (- 2 i)$ .

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Step 5.2.2.2.1

Multiply $- 2$ by $2$ .

$y = x^{2} - 4 i i$

Step 5.2.2.2.2

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

$y = x^{2} - 4 (i i)$

Step 5.2.2.2.3

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

$y = x^{2} - 4 (i i)$

Step 5.2.2.2.4

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

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

Step 5.2.2.2.5

Add $1$ and $1$ .

$y = x^{2} - 4 i^{2}$

Step 5.2.2.3

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

$y = x^{2} - 4 \cdot - 1$

Step 5.2.2.4

Multiply $- 4$ by $- 1$ .

$y = x^{2} + 4$

Step 6