Find the Equation Given the Roots 2i

$2 i$

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

$y = 0$ at the roots

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

The factor is $x - 2 i$

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

The factor is $x + 2 i$

Combine all the factors into a single equation.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify terms.

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

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

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

Subtract $2 i x$ from $2 i x$ .

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

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

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- 2 i (2 i)$ .

Multiply $2$ by $- 2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

$y = x^{2} + 4$