Precalculus Examples

$- 6 \sqrt{3} - 6 i$

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 $x = - 6 \sqrt{3} - 6 i$ was found by solving for $x$ when $x - (- 6 \sqrt{3} - 6 i) = y$ and $y = 0$ .

The factor is $x + 6 \sqrt{3} + 6 i$

Step 3

The root at $x = - 6 \sqrt{3} + 6 i$ was found by solving for $x$ when $x - (- 6 \sqrt{3} + 6 i) = y$ and $y = 0$ .

The factor is $x + 6 \sqrt{3} - 6 i$

Step 4

Combine all the factors into a single equation.

$y = (x + 6 \sqrt{3} + 6 i) (x + 6 \sqrt{3} - 6 i)$

Step 5

Multiply all the factors to simplify the equation $y = (x + 6 \sqrt{3} + 6 i) (x + 6 \sqrt{3} - 6 i)$ .

Tap for more steps...

Step 5.1

Expand $(x + 6 \sqrt{3} + 6 i) (x + 6 \sqrt{3} - 6 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x (6 \sqrt{3}) + x (- 6 i) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 \sqrt{3} (- 6 i) + 6 i x + 6 i (6 \sqrt{3}) + 6 i (- 6 i)$

Step 5.2

Simplify terms.

Tap for more steps...

Step 5.2.1

Combine the opposite terms in $x \cdot x + x (6 \sqrt{3}) + x (- 6 i) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 \sqrt{3} (- 6 i) + 6 i x + 6 i (6 \sqrt{3}) + 6 i (- 6 i)$ .

Tap for more steps...

Step 5.2.1.1

Reorder the factors in the terms $x (- 6 i)$ and $6 i x$ .

$y = x \cdot x + x (6 \sqrt{3}) - 6 i x + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 \sqrt{3} (- 6 i) + 6 i x + 6 i (6 \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.1.2

Add $- 6 i x$ and $6 i x$ .

$y = x \cdot x + x (6 \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 \sqrt{3} (- 6 i) + 0 + 6 i (6 \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.1.3

Add $x \cdot x + x (6 \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 \sqrt{3} (- 6 i)$ and $0$ .

$y = x \cdot x + x (6 \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 \sqrt{3} (- 6 i) + 6 i (6 \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.1.4

Reorder the factors in the terms $6 \sqrt{3} (- 6 i)$ and $6 i (6 \sqrt{3})$ .

$y = x \cdot x + x (6 \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) - 6 \cdot (6 i \sqrt{3}) + 6 \cdot (6 i \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.1.5

Add $- 6 \cdot 6 i \sqrt{3}$ and $6 \cdot 6 i \sqrt{3}$ .

$y = x \cdot x + x (6 \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 0 + 6 i (- 6 i)$

Step 5.2.1.6

Add $x \cdot x + x (6 \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3})$ and $0$ .

$y = x \cdot x + x (6 \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

Multiply $x$ by $x$ .

$y = x^{2} + x (6 \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.2.2

Move $6$ to the left of $x$ .

$y = x^{2} + 6 \cdot (x \sqrt{3}) + 6 \sqrt{3} x + 6 \sqrt{3} (6 \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.2.3

Multiply $6 \sqrt{3} (6 \sqrt{3})$ .

Tap for more steps...

Step 5.2.2.3.1

Multiply $6$ by $6$ .

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 \sqrt{3} \sqrt{3} + 6 i (- 6 i)$

Step 5.2.2.3.2

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 (\sqrt{3} \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.2.3.3

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 (\sqrt{3} \sqrt{3}) + 6 i (- 6 i)$

Step 5.2.2.3.4

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 {\sqrt{3}}^{1 + 1} + 6 i (- 6 i)$

Step 5.2.2.3.5

Add $1$ and $1$ .

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 {\sqrt{3}}^{2} + 6 i (- 6 i)$

Step 5.2.2.4

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

Tap for more steps...

Step 5.2.2.4.1

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 {(3^{\frac{1}{2}})}^{2} + 6 i (- 6 i)$

Step 5.2.2.4.2

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 \cdot 3^{\frac{1}{2} \cdot 2} + 6 i (- 6 i)$

Step 5.2.2.4.3

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 \cdot 3^{\frac{2}{2}} + 6 i (- 6 i)$

Step 5.2.2.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.2.4.4.1

Cancel the common factor.

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 \cdot 3^{\frac{2}{2}} + 6 i (- 6 i)$

Step 5.2.2.4.4.2

Rewrite the expression.

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 \cdot 3 + 6 i (- 6 i)$

Step 5.2.2.4.5

Evaluate the exponent.

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 36 \cdot 3 + 6 i (- 6 i)$

Step 5.2.2.5

Multiply $36$ by $3$ .

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 108 + 6 i (- 6 i)$

Step 5.2.2.6

Multiply $6 i (- 6 i)$ .

Tap for more steps...

Step 5.2.2.6.1

Multiply $- 6$ by $6$ .

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 108 - 36 i i$

Step 5.2.2.6.2

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 108 - 36 (i i)$

Step 5.2.2.6.3

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 108 - 36 (i i)$

Step 5.2.2.6.4

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 108 - 36 i^{1 + 1}$

Step 5.2.2.6.5

Add $1$ and $1$ .

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 108 - 36 i^{2}$

Step 5.2.2.7

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

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 108 - 36 \cdot - 1$

Step 5.2.2.8

Multiply $- 36$ by $- 1$ .

$y = x^{2} + 6 x \sqrt{3} + 6 \sqrt{3} x + 108 + 36$

Step 5.2.3

Simplify by adding terms.

Tap for more steps...

Step 5.2.3.1

Reorder the factors of $6 \sqrt{3} x$ .

$y = x^{2} + 6 x \sqrt{3} + 6 x \sqrt{3} + 108 + 36$

Step 5.2.3.2

Add $6 x \sqrt{3}$ and $6 x \sqrt{3}$ .

$y = x^{2} + 12 x \sqrt{3} + 108 + 36$

Step 5.2.3.3

Add $108$ and $36$ .

$y = x^{2} + 12 x \sqrt{3} + 144$

Step 6