Find the Equation Given the Roots -3i square root of 2

$- 3 i \sqrt{2}$

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 $x = - 3 i \sqrt{2}$ was found by solving for $x$ when $x - (- 3 i \sqrt{2}) = y$ and $y = 0$ .

The factor is $x + 3 i \sqrt{2}$

Step 3

The root at $x = 3 i \sqrt{2}$ was found by solving for $x$ when $x - (3 i \sqrt{2}) = y$ and $y = 0$ .

The factor is $x - 3 i \sqrt{2}$

Step 4

Combine all the factors into a single equation.

$y = (x + 3 i \sqrt{2}) (x - 3 i \sqrt{2})$

Step 5

Multiply all the factors to simplify the equation $y = (x + 3 i \sqrt{2}) (x - 3 i \sqrt{2})$ .

Expand $(x + 3 i \sqrt{2}) (x - 3 i \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$y = x (x - 3 i \sqrt{2}) + 3 i \sqrt{2} (x - 3 i \sqrt{2})$

Apply the distributive property.

$y = x \cdot x + x (- 3 i \sqrt{2}) + 3 i \sqrt{2} (x - 3 i \sqrt{2})$

Apply the distributive property.

$y = x \cdot x + x (- 3 i \sqrt{2}) + 3 i \sqrt{2} x + 3 i \sqrt{2} (- 3 i \sqrt{2})$

Simplify terms.

Combine the opposite terms in $x \cdot x + x (- 3 i \sqrt{2}) + 3 i \sqrt{2} x + 3 i \sqrt{2} (- 3 i \sqrt{2})$ .

Reorder the factors in the terms $x (- 3 i \sqrt{2})$ and $3 i \sqrt{2} x$ .

$y = x \cdot x - 3 i x \sqrt{2} + 3 i x \sqrt{2} + 3 i \sqrt{2} (- 3 i \sqrt{2})$

Add $- 3 i x \sqrt{2}$ and $3 i x \sqrt{2}$ .

$y = x \cdot x + 0 + 3 i \sqrt{2} (- 3 i \sqrt{2})$

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

$y = x \cdot x + 3 i \sqrt{2} (- 3 i \sqrt{2})$

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + 3 i \sqrt{2} (- 3 i \sqrt{2})$

Multiply $3 i \sqrt{2} (- 3 i \sqrt{2})$ .

Multiply $- 3$ by $3$ .

$y = x^{2} - 9 i \sqrt{2} (i \sqrt{2})$

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

$y = x^{2} - 9 (i i) \sqrt{2} \sqrt{2}$

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

$y = x^{2} - 9 (i i) \sqrt{2} \sqrt{2}$

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

$y = x^{2} - 9 i^{1 + 1} \sqrt{2} \sqrt{2}$

Add $1$ and $1$ .

$y = x^{2} - 9 i^{2} \sqrt{2} \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$y = x^{2} - 9 i^{2} (\sqrt{2} \sqrt{2})$

Raise $\sqrt{2}$ to the power of $1$ .

$y = x^{2} - 9 i^{2} (\sqrt{2} \sqrt{2})$

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

$y = x^{2} - 9 i^{2} {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$y = x^{2} - 9 i^{2} {\sqrt{2}}^{2}$

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

$y = x^{2} - 9 \cdot (- 1 {\sqrt{2}}^{2})$

Multiply $- 9$ by $- 1$ .

$y = x^{2} + 9 {\sqrt{2}}^{2}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y = x^{2} + 9 {(2^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = x^{2} + 9 \cdot 2^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$y = x^{2} + 9 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = x^{2} + 9 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$y = x^{2} + 9 \cdot 2$

Evaluate the exponent.

$y = x^{2} + 9 \cdot 2$

Multiply $9$ by $2$ .

$y = x^{2} + 18$

Step 6

Find the Equation Given the Roots -3i square root of 2

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