Find the Equation Given the Roots 14+ square root of 13 , 3+2i

$14 + \sqrt{13}$ , $3 + 2 i$

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 $x = 14 + \sqrt{13}$ was found by solving for $x$ when $x - (14 + \sqrt{13}) = y$ and $y = 0$ .

The factor is $x - 14 - \sqrt{13}$

Step 3

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

The factor is $x - 3 - 2 i$

Step 4

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

The factor is $x - 3 + 2 i$

Step 5

Combine all the factors into a single equation.

$y = (x - 14 - \sqrt{13}) (x - 3 - 2 i) (x - 3 + 2 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 14 - \sqrt{13}) (x - 3 - 2 i) (x - 3 + 2 i)$ .

Expand $(x - 14 - \sqrt{13}) (x - 3 - 2 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 3 + x (- 2 i) - 14 x - 14 \cdot - 3 - 14 (- 2 i) - \sqrt{13} x - \sqrt{13} \cdot - 3 - \sqrt{13} (- 2 i)) (x - 3 + 2 i)$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 3 + x (- 2 i) - 14 x - 14 \cdot - 3 - 14 (- 2 i) - \sqrt{13} x - \sqrt{13} \cdot - 3 - \sqrt{13} (- 2 i)) (x - 3 + 2 i)$

Move $- 3$ to the left of $x$ .

$y = (x^{2} - 3 \cdot x + x (- 2 i) - 14 x - 14 \cdot - 3 - 14 (- 2 i) - \sqrt{13} x - \sqrt{13} \cdot - 3 - \sqrt{13} (- 2 i)) (x - 3 + 2 i)$

Multiply $- 14$ by $- 3$ .

$y = (x^{2} - 3 x + x (- 2 i) - 14 x + 42 - 14 (- 2 i) - \sqrt{13} x - \sqrt{13} \cdot - 3 - \sqrt{13} (- 2 i)) (x - 3 + 2 i)$

Multiply $- 2$ by $- 14$ .

$y = (x^{2} - 3 x + x (- 2 i) - 14 x + 42 + 28 i - \sqrt{13} x - \sqrt{13} \cdot - 3 - \sqrt{13} (- 2 i)) (x - 3 + 2 i)$

Multiply $- 3$ by $- 1$ .

$y = (x^{2} - 3 x + x (- 2 i) - 14 x + 42 + 28 i - \sqrt{13} x + 3 \sqrt{13} - \sqrt{13} (- 2 i)) (x - 3 + 2 i)$

Multiply $- 2$ by $- 1$ .

$y = (x^{2} - 3 x + x (- 2 i) - 14 x + 42 + 28 i - \sqrt{13} x + 3 \sqrt{13} + 2 \sqrt{13} i) (x - 3 + 2 i)$

Subtract $14 x$ from $- 3 x$ .

$y = (x^{2} + x (- 2 i) - 17 x + 42 + 28 i - \sqrt{13} x + 3 \sqrt{13} + 2 \sqrt{13} i) (x - 3 + 2 i)$

Expand $(x^{2} + x (- 2 i) - 17 x + 42 + 28 i - \sqrt{13} x + 3 \sqrt{13} + 2 \sqrt{13} i) (x - 3 + 2 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 3 + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Simplify terms.

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

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

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

$y = x^{2 + 1} + x^{2} \cdot - 3 + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 3 + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Move $- 3$ to the left of $x^{2}$ .

$y = x^{3} - 3 \cdot x^{2} + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Move $2$ to the left of $x^{2}$ .

$y = x^{3} - 3 x^{2} + 2 \cdot (x^{2} i) + x (- 2 i) x + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x \cdot x (- 2 i) + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $x$ by $x$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + x (- 2 i) \cdot - 3 + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $- 3$ by $- 2$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + x (6 i) + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

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

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 \cdot (x i) + x (- 2 i) (2 i) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $x (- 2 i) (2 i)$ .

Multiply $2$ by $- 2$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 i) i - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

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

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 (i i)) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

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

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

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 i^{1 + 1}) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Add $1$ and $1$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 i^{2}) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

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

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x (- 4 \cdot - 1) - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $- 4$ by $- 1$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + x \cdot 4 - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Move $4$ to the left of $x$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 \cdot x - 17 x \cdot x - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 (x \cdot x) - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $x$ by $x$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} - 17 x \cdot - 3 - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $- 3$ by $- 17$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 17 x (2 i) + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $2$ by $- 17$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x + 42 \cdot - 3 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $42$ by $- 3$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 42 (2 i) + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $2$ by $42$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x + 28 i \cdot - 3 + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $- 3$ by $28$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i + 28 i (2 i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $28 i (2 i)$ .

Multiply $2$ by $28$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i + 56 i i - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

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

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i + 56 (i i) - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

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

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

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i + 56 i^{1 + 1} - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Add $1$ and $1$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i + 56 i^{2} - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

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

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i + 56 \cdot - 1 - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $56$ by $- 1$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x \cdot x - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} (x \cdot x) - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $x$ by $x$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} - \sqrt{13} x \cdot - 3 - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $- 3$ by $- 1$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - \sqrt{13} x (2 i) + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $2$ by $- 1$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x + 3 \sqrt{13} \cdot - 3 + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $- 3$ by $3$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x - 9 \sqrt{13} + 3 \sqrt{13} (2 i) + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $2$ by $3$ .

$y = x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 2 \sqrt{13} i x + 2 \sqrt{13} i \cdot - 3 + 2 \sqrt{13} i (2 i)$

Multiply $- 3$ by $2$ .

Multiply $2 \sqrt{13} i (2 i)$ .

Multiply $2$ by $2$ .

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

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

Add $1$ and $1$ .

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

Multiply $- 1$ by $4$ .

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 3 x^{2} + 2 x^{2} i + x^{2} (- 2 i) + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 2 \sqrt{13} i x - 6 \sqrt{13} i - 4 \sqrt{13}$ .

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

$y = x^{3} - 3 x^{2} + 2 i x^{2} - 2 i x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 2 \sqrt{13} i x - 6 \sqrt{13} i - 4 \sqrt{13}$

Subtract $2 i x^{2}$ from $2 i x^{2}$ .

$y = x^{3} - 3 x^{2} + 0 + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 2 \sqrt{13} i x - 6 \sqrt{13} i - 4 \sqrt{13}$

Add $x^{3} - 3 x^{2}$ and $0$ .

$y = x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 84 i + 28 i x - 84 i - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 2 \sqrt{13} i x - 6 \sqrt{13} i - 4 \sqrt{13}$

Subtract $84 i$ from $84 i$ .

$y = x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x + 0 - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 2 \sqrt{13} i x - 6 \sqrt{13} i - 4 \sqrt{13}$

Add $x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x$ and $0$ .

$y = x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 \sqrt{13} x i + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 2 \sqrt{13} i x - 6 \sqrt{13} i - 4 \sqrt{13}$

Reorder the factors in the terms $- 2 \sqrt{13} x i$ and $2 \sqrt{13} i x$ .

$y = x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x - 2 i x \sqrt{13} + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 2 i x \sqrt{13} - 6 \sqrt{13} i - 4 \sqrt{13}$

Add $- 2 i x \sqrt{13}$ and $2 i x \sqrt{13}$ .

$y = x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i + 0 - 6 \sqrt{13} i - 4 \sqrt{13}$

Add $x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i$ and $0$ .

$y = x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} + 6 \sqrt{13} i - 6 \sqrt{13} i - 4 \sqrt{13}$

Subtract $6 \sqrt{13} i$ from $6 \sqrt{13} i$ .

$y = x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} + 0 - 4 \sqrt{13}$

Add $x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13}$ and $0$ .

$y = x^{3} - 3 x^{2} + 6 x i + 4 x - 17 x^{2} + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Subtract $17 x^{2}$ from $- 3 x^{2}$ .

$y = x^{3} - 20 x^{2} + 6 x i + 4 x + 51 x - 34 x i + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Subtract $34 x i$ from $6 x i$ .

$y = x^{3} - 20 x^{2} - 28 x i + 4 x + 51 x + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Combine the opposite terms in $x^{3} - 20 x^{2} - 28 x i + 4 x + 51 x + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$ .

Reorder the factors in the terms $- 28 x i$ and $28 i x$ .

$y = x^{3} - 20 x^{2} - 28 i x + 4 x + 51 x + 42 x - 126 + 28 i x - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Add $- 28 i x$ and $28 i x$ .

$y = x^{3} - 20 x^{2} + 4 x + 51 x + 42 x - 126 + 0 - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Add $x^{3} - 20 x^{2} + 4 x + 51 x + 42 x - 126$ and $0$ .

$y = x^{3} - 20 x^{2} + 4 x + 51 x + 42 x - 126 - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Add $4 x$ and $51 x$ .

$y = x^{3} - 20 x^{2} + 55 x + 42 x - 126 - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Add $55 x$ and $42 x$ .

$y = x^{3} - 20 x^{2} + 97 x - 126 - 56 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Subtract $56$ from $- 126$ .

$y = x^{3} - 20 x^{2} + 97 x - 182 - \sqrt{13} x^{2} + 3 \sqrt{13} x + 3 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Add $3 \sqrt{13} x$ and $3 \sqrt{13} x$ .

$y = x^{3} - 20 x^{2} + 97 x - 182 - \sqrt{13} x^{2} + 6 \sqrt{13} x - 9 \sqrt{13} - 4 \sqrt{13}$

Subtract $4 \sqrt{13}$ from $- 9 \sqrt{13}$ .

$y = x^{3} - 20 x^{2} + 97 x - 182 - \sqrt{13} x^{2} + 6 \sqrt{13} x - 13 \sqrt{13}$

Reorder.

Move $- 182$ .

$y = x^{3} - 20 x^{2} + 97 x - \sqrt{13} x^{2} + 6 \sqrt{13} x - 182 - 13 \sqrt{13}$

Move $97 x$ .

$y = x^{3} - 20 x^{2} - \sqrt{13} x^{2} + 97 x + 6 \sqrt{13} x - 182 - 13 \sqrt{13}$

Step 7

Find the Equation Given the Roots 14+ square root of 13 , 3+2i

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