Precalculus Examples

$- 36 + 36 \sqrt{3 i}$

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 $x = - 36 + 36 \sqrt{3 i}$ was found by solving for $x$ when $x - (- 36 + 36 \sqrt{3 i}) = y$ and $y = 0$ .

The factor is $x + 36 - 36 \sqrt{3 i}$

Step 3

The root at $x = - 36 - 36 \sqrt{3 i}$ was found by solving for $x$ when $x - (- 36 - 36 \sqrt{3 i}) = y$ and $y = 0$ .

The factor is $x + 36 + 36 \sqrt{3 i}$

Step 4

Combine all the factors into a single equation.

$y = (x + 36 - 36 \sqrt{3 i}) (x + 36 + 36 \sqrt{3 i})$

Step 5

Multiply all the factors to simplify the equation $y = (x + 36 - 36 \sqrt{3 i}) (x + 36 + 36 \sqrt{3 i})$ .

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Step 5.1

Expand $(x + 36 - 36 \sqrt{3 i}) (x + 36 + 36 \sqrt{3 i})$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot 36 + x (36 \sqrt{3 i}) + 36 x + 36 \cdot 36 + 36 (36 \sqrt{3 i}) - 36 \sqrt{3 i} x - 36 \sqrt{3 i} \cdot 36 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2

Simplify terms.

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Step 5.2.1

Combine the opposite terms in $x \cdot x + x \cdot 36 + x (36 \sqrt{3 i}) + 36 x + 36 \cdot 36 + 36 (36 \sqrt{3 i}) - 36 \sqrt{3 i} x - 36 \sqrt{3 i} \cdot 36 - 36 \sqrt{3 i} (36 \sqrt{3 i})$ .

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Step 5.2.1.1

Reorder the factors in the terms $x (36 \sqrt{3 i})$ and $- 36 \sqrt{3 i} x$ .

$y = x \cdot x + x \cdot 36 + 36 x \sqrt{3 i} + 36 x + 36 \cdot 36 + 36 (36 \sqrt{3 i}) - 36 x \sqrt{3 i} - 36 \sqrt{3 i} \cdot 36 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.1.2

Subtract $36 x \sqrt{3 i}$ from $36 x \sqrt{3 i}$ .

$y = x \cdot x + x \cdot 36 + 36 x + 36 \cdot 36 + 36 (36 \sqrt{3 i}) + 0 - 36 \sqrt{3 i} \cdot 36 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.1.3

Add $x \cdot x + x \cdot 36 + 36 x + 36 \cdot 36 + 36 (36 \sqrt{3 i})$ and $0$ .

$y = x \cdot x + x \cdot 36 + 36 x + 36 \cdot 36 + 36 (36 \sqrt{3 i}) - 36 \sqrt{3 i} \cdot 36 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.1.4

Reorder the factors in the terms $36 (36 \sqrt{3 i})$ and $- 36 \sqrt{3 i} \cdot 36$ .

$y = x \cdot x + x \cdot 36 + 36 x + 36 \cdot 36 + 36 \cdot (36 \sqrt{3 i}) - 36 \cdot (36 \sqrt{3 i}) - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.1.5

Subtract $36 \cdot 36 \sqrt{3 i}$ from $36 \cdot 36 \sqrt{3 i}$ .

$y = x \cdot x + x \cdot 36 + 36 x + 36 \cdot 36 + 0 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.1.6

Add $x \cdot x + x \cdot 36 + 36 x + 36 \cdot 36$ and $0$ .

$y = x \cdot x + x \cdot 36 + 36 x + 36 \cdot 36 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot 36 + 36 x + 36 \cdot 36 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.2.2

Move $36$ to the left of $x$ .

$y = x^{2} + 36 \cdot x + 36 x + 36 \cdot 36 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.2.3

Multiply $36$ by $36$ .

$y = x^{2} + 36 x + 36 x + 1296 - 36 \sqrt{3 i} (36 \sqrt{3 i})$

Step 5.2.2.4

Multiply $- 36 \sqrt{3 i} (36 \sqrt{3 i})$ .

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Step 5.2.2.4.1

Multiply $36$ by $- 36$ .

$y = x^{2} + 36 x + 36 x + 1296 - 1296 \sqrt{3 i} \sqrt{3 i}$

Step 5.2.2.4.2

Raise $\sqrt{3 i}$ to the power of $1$ .

$y = x^{2} + 36 x + 36 x + 1296 - 1296 (\sqrt{3 i} \sqrt{3 i})$

Step 5.2.2.4.3

Raise $\sqrt{3 i}$ to the power of $1$ .

$y = x^{2} + 36 x + 36 x + 1296 - 1296 (\sqrt{3 i} \sqrt{3 i})$

Step 5.2.2.4.4

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

$y = x^{2} + 36 x + 36 x + 1296 - 1296 {\sqrt{3 i}}^{1 + 1}$

Step 5.2.2.4.5

Add $1$ and $1$ .

$y = x^{2} + 36 x + 36 x + 1296 - 1296 {\sqrt{3 i}}^{2}$

Step 5.2.2.5

Rewrite ${\sqrt{3 i}}^{2}$ as $3 i$ .

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Step 5.2.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 i}$ as ${(3 i)}^{\frac{1}{2}}$ .

$y = x^{2} + 36 x + 36 x + 1296 - 1296 {({(3 i)}^{\frac{1}{2}})}^{2}$

Step 5.2.2.5.2

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

$y = x^{2} + 36 x + 36 x + 1296 - 1296 {(3 i)}^{\frac{1}{2} \cdot 2}$

Step 5.2.2.5.3

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

$y = x^{2} + 36 x + 36 x + 1296 - 1296 {(3 i)}^{\frac{2}{2}}$

Step 5.2.2.5.4

Cancel the common factor of $2$ .

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Step 5.2.2.5.4.1

Cancel the common factor.

$y = x^{2} + 36 x + 36 x + 1296 - 1296 {(3 i)}^{\frac{2}{2}}$

Step 5.2.2.5.4.2

Rewrite the expression.

$y = x^{2} + 36 x + 36 x + 1296 - 1296 (3 i)$

Step 5.2.2.5.5

Simplify.

$y = x^{2} + 36 x + 36 x + 1296 - 1296 (3 i)$

Step 5.2.2.6

Multiply $3$ by $- 1296$ .

$y = x^{2} + 36 x + 36 x + 1296 - 3888 i$

Step 5.2.3

Add $36 x$ and $36 x$ .

$y = x^{2} + 72 x + 1296 - 3888 i$

Step 6