Find the Equation Given the Roots 5+6i

$5 + 6 i$

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 $x = 5 + 6 i$ was found by solving for $x$ when $x - (5 + 6 i) = y$ and $y = 0$ .

The factor is $x - 5 - 6 i$

Step 3

The root at $x = 5 - 6 i$ was found by solving for $x$ when $x - (5 - 6 i) = y$ and $y = 0$ .

The factor is $x - 5 + 6 i$

Step 4

Combine all the factors into a single equation.

$y = (x - 5 - 6 i) (x - 5 + 6 i)$

Step 5

Multiply all the factors to simplify the equation $y = (x - 5 - 6 i) (x - 5 + 6 i)$ .

Expand $(x - 5 - 6 i) (x - 5 + 6 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot - 5 + x (6 i) - 5 x - 5 \cdot - 5 - 5 (6 i) - 6 i x - 6 i \cdot - 5 - 6 i (6 i)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 5 + x (6 i) - 5 x - 5 \cdot - 5 - 5 (6 i) - 6 i x - 6 i \cdot - 5 - 6 i (6 i)$ .

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

$y = x \cdot x + x \cdot - 5 + 6 i x - 5 x - 5 \cdot - 5 - 5 (6 i) - 6 i x - 6 i \cdot - 5 - 6 i (6 i)$

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

$y = x \cdot x + x \cdot - 5 + 0 - 5 x - 5 \cdot - 5 - 5 (6 i) - 6 i \cdot - 5 - 6 i (6 i)$

Add $x \cdot x + x \cdot - 5$ and $0$ .

$y = x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (6 i) - 6 i \cdot - 5 - 6 i (6 i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (6 i) - 6 i \cdot - 5 - 6 i (6 i)$

Move $- 5$ to the left of $x$ .

$y = x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - 5 (6 i) - 6 i \cdot - 5 - 6 i (6 i)$

Multiply $- 5$ by $- 5$ .

$y = x^{2} - 5 x - 5 x + 25 - 5 (6 i) - 6 i \cdot - 5 - 6 i (6 i)$

Multiply $6$ by $- 5$ .

$y = x^{2} - 5 x - 5 x + 25 - 30 i - 6 i \cdot - 5 - 6 i (6 i)$

Multiply $- 5$ by $- 6$ .

$y = x^{2} - 5 x - 5 x + 25 - 30 i + 30 i - 6 i (6 i)$

Multiply $- 6 i (6 i)$ .

Multiply $6$ by $- 6$ .

$y = x^{2} - 5 x - 5 x + 25 - 30 i + 30 i - 36 i i$

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

$y = x^{2} - 5 x - 5 x + 25 - 30 i + 30 i - 36 (i i)$

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

$y = x^{2} - 5 x - 5 x + 25 - 30 i + 30 i - 36 (i i)$

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

$y = x^{2} - 5 x - 5 x + 25 - 30 i + 30 i - 36 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{2} - 5 x - 5 x + 25 - 30 i + 30 i - 36 i^{2}$

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

$y = x^{2} - 5 x - 5 x + 25 - 30 i + 30 i - 36 \cdot - 1$

Multiply $- 36$ by $- 1$ .

$y = x^{2} - 5 x - 5 x + 25 - 30 i + 30 i + 36$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 5 x - 5 x + 25 - 30 i + 30 i + 36$ .

Add $- 30 i$ and $30 i$ .

$y = x^{2} - 5 x - 5 x + 25 + 0 + 36$

Add $x^{2} - 5 x - 5 x + 25$ and $0$ .

$y = x^{2} - 5 x - 5 x + 25 + 36$

Subtract $5 x$ from $- 5 x$ .

$y = x^{2} - 10 x + 25 + 36$

Add $25$ and $36$ .

$y = x^{2} - 10 x + 61$

Step 6

Find the Equation Given the Roots 5+6i

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