Find the Quadratic Equation -3i , 3i

$- 3 i$ , $3 i$

Step 1

$x = - 3 i$ and $x = 3 i$ are the two real distinct solutions for the quadratic equation, which means that $x - (- 3 i)$ and $x - 3 i$ are the factors of the quadratic equation.

$(x + 3 i) (x - 3 i) = 0$

Step 2

Expand $(x + 3 i) (x - 3 i)$ using the FOIL Method.

Apply the distributive property.

$x (x - 3 i) + 3 i (x - 3 i) = 0$

Apply the distributive property.

$x \cdot x + x (- 3 i) + 3 i (x - 3 i) = 0$

Apply the distributive property.

$x \cdot x + x (- 3 i) + 3 i x + 3 i (- 3 i) = 0$

Step 3

Simplify terms.

Combine the opposite terms in $x \cdot x + x (- 3 i) + 3 i x + 3 i (- 3 i)$ .

Reorder the factors in the terms $x (- 3 i)$ and $3 i x$ .

$x \cdot x - 3 i x + 3 i x + 3 i (- 3 i) = 0$

Add $- 3 i x$ and $3 i x$ .

$x \cdot x + 0 + 3 i (- 3 i) = 0$

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

$x \cdot x + 3 i (- 3 i) = 0$

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + 3 i (- 3 i) = 0$

Multiply $3 i (- 3 i)$ .

Multiply $- 3$ by $3$ .

$x^{2} - 9 i i = 0$

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

$x^{2} - 9 (i i) = 0$

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

$x^{2} - 9 (i i) = 0$

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

$x^{2} - 9 i^{1 + 1} = 0$

Add $1$ and $1$ .

$x^{2} - 9 i^{2} = 0$

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

$x^{2} - 9 \cdot - 1 = 0$

Multiply $- 9$ by $- 1$ .

$x^{2} + 9 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${- 3 i, 3 i}$ is $y = x^{2} + 9$ .

$y = x^{2} + 9$

Step 5

Find the Quadratic Equation -3i , 3i

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