Find the Equation Given the Roots ( plus or minus 4)i

$(\pm 4) i$

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 $x = i (\pm 4)$ was found by solving for $x$ when $x - (i (\pm 4)) = y$ and $y = 0$ .

The factor is $x - i (\pm 4)$

Step 3

The root at $x = - i (\pm 4)$ was found by solving for $x$ when $x - (- i (\pm 4)) = y$ and $y = 0$ .

The factor is $x + i (\pm 4)$

Step 4

Combine all the factors into a single equation.

$y = (x - i (\pm 4)) (x + i (\pm 4))$

Step 5

Multiply all the factors to simplify the equation $y = (x - i (\pm 4)) (x + i (\pm 4))$ .

Expand $(x - i (\pm 4)) (x + i (\pm 4))$ using the FOIL Method.

Apply the distributive property.

$y = x (x + i (\pm 4)) - i (\pm 4) (x + i (\pm 4))$

Apply the distributive property.

$y = x \cdot x + x (i (\pm 4)) - i (\pm 4) (x + i (\pm 4))$

Apply the distributive property.

$y = x \cdot x + x (i (\pm 4)) - i (\pm 4) x - i (\pm 4) (i (\pm 4))$

Simplify terms.

Combine the opposite terms in $x \cdot x + x (i (\pm 4)) - i (\pm 4) x - i (\pm 4) (i (\pm 4))$ .

Reorder the factors in the terms $x (i (\pm 4))$ and $- i (\pm 4) x$ .

$y = x \cdot x + i x (\pm 4) - i x (\pm 4) - i (\pm 4) (i (\pm 4))$

Subtract $i x (\pm 4)$ from $i x (\pm 4)$ .

$y = x \cdot x + 0 - i (\pm 4) (i (\pm 4))$

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

$y = x \cdot x - i (\pm 4) (i (\pm 4))$

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- i (\pm 4) (i (\pm 4))$ .

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

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

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

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

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

$y = x^{2} - i^{1 + 1} (\pm 4) (\pm 4)$

Add $1$ and $1$ .

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

Raise $\pm 4$ to the power of $1$ .

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

Raise $\pm 4$ to the power of $1$ .

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

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

$y = x^{2} - i^{2} {(\pm 4)}^{1 + 1}$

Add $1$ and $1$ .

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

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

$y = x^{2} + 1 {(\pm 4)}^{2}$

Multiply $- 1$ by $- 1$ .

$y = x^{2} + 1 {(\pm 4)}^{2}$

Multiply ${(\pm 4)}^{2}$ by $1$ .

$y = x^{2} + {(\pm 4)}^{2}$

Remove the plus-minus sign on $\pm 4$ because it is raised to an even power.

$y = x^{2} + 4^{2}$

Raise $4$ to the power of $2$ .

$y = x^{2} + 16$

Step 6

Find the Equation Given the Roots ( plus or minus 4)i

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