Find the Equation Given the Roots -2 , 2+3i

$- 2$ , $2 + 3 i$

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 $x = - 2$ was found by solving for $x$ when $x - (- 2) = y$ and $y = 0$ .

The factor is $x + 2$

Step 3

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

The factor is $x - 2 - 3 i$

Step 4

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

The factor is $x - 2 + 3 i$

Step 5

Combine all the factors into a single equation.

$y = (x + 2) (x - 2 - 3 i) (x - 2 + 3 i)$

Step 6

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

Expand $(x + 2) (x - 2 - 3 i)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

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

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

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

Add $- 2 x$ and $2 x$ .

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

Add $x \cdot x + x (- 3 i)$ and $0$ .

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $2$ by $- 2$ .

$y = (x^{2} + x (- 3 i) - 4 + 2 (- 3 i)) (x - 2 + 3 i)$

Multiply $- 3$ by $2$ .

$y = (x^{2} + x (- 3 i) - 4 - 6 i) (x - 2 + 3 i)$

Expand $(x^{2} + x (- 3 i) - 4 - 6 i) (x - 2 + 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 2 + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 2 + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 2 + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 2 + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

$y = x^{2 + 1} + x^{2} \cdot - 2 + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 2 + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 2 + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 2 + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

$y = x^{3} - 2 \cdot x^{2} + x^{2} (3 i) + x (- 3 i) x + x (- 3 i) \cdot - 2 + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

$y = x^{3} - 2 x^{2} + 3 \cdot (x^{2} i) + x (- 3 i) x + x (- 3 i) \cdot - 2 + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

Move $x$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x \cdot x (- 3 i) + x (- 3 i) \cdot - 2 + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Multiply $x$ by $x$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + x (- 3 i) \cdot - 2 + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Multiply $- 2$ by $- 3$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + x (6 i) + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 \cdot (x i) + x (- 3 i) (3 i) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Multiply $x (- 3 i) (3 i)$ .

Multiply $3$ by $- 3$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + x (- 9 i) i - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + x (- 9 (i i)) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + x (- 9 (i i)) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + x (- 9 i^{1 + 1}) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Add $1$ and $1$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + x (- 9 i^{2}) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + x (- 9 \cdot - 1) - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Multiply $- 9$ by $- 1$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + x \cdot 9 - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Move $9$ to the left of $x$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 \cdot x - 4 x - 4 \cdot - 2 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Multiply $- 4$ by $- 2$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 4 (3 i) - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Multiply $3$ by $- 4$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x - 6 i \cdot - 2 - 6 i (3 i)$

Multiply $- 2$ by $- 6$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i - 6 i (3 i)$

Multiply $- 6 i (3 i)$ .

Multiply $3$ by $- 6$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i - 18 i i$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i - 18 (i i)$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i - 18 (i i)$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i - 18 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i - 18 i^{2}$

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

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i - 18 \cdot - 1$

Multiply $- 18$ by $- 1$ .

$y = x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i + 18$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 2 x^{2} + 3 x^{2} i + x^{2} (- 3 i) + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i + 18$ .

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

$y = x^{3} - 2 x^{2} + 3 i x^{2} - 3 i x^{2} + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i + 18$

Subtract $3 i x^{2}$ from $3 i x^{2}$ .

$y = x^{3} - 2 x^{2} + 0 + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i + 18$

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

$y = x^{3} - 2 x^{2} + 6 x i + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i + 18$

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

$y = x^{3} - 2 x^{2} + 6 i x + 9 x - 4 x + 8 - 12 i - 6 i x + 12 i + 18$

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

$y = x^{3} - 2 x^{2} + 9 x - 4 x + 8 - 12 i + 0 + 12 i + 18$

Add $x^{3} - 2 x^{2} + 9 x - 4 x + 8 - 12 i$ and $0$ .

$y = x^{3} - 2 x^{2} + 9 x - 4 x + 8 - 12 i + 12 i + 18$

Add $- 12 i$ and $12 i$ .

$y = x^{3} - 2 x^{2} + 9 x - 4 x + 8 + 0 + 18$

Add $x^{3} - 2 x^{2} + 9 x - 4 x + 8$ and $0$ .

$y = x^{3} - 2 x^{2} + 9 x - 4 x + 8 + 18$

Subtract $4 x$ from $9 x$ .

$y = x^{3} - 2 x^{2} + 5 x + 8 + 18$

Add $8$ and $18$ .

$y = x^{3} - 2 x^{2} + 5 x + 26$

Step 7

Find the Equation Given the Roots -2 , 2+3i

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