Find the Equation Given the Roots 1-6i

$1 - 6 i$

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 $x = 1 - 6 i$ was found by solving for $x$ when $x - (1 - 6 i) = y$ and $y = 0$ .

The factor is $x - 1 + 6 i$

Step 3

The root at $x = 1 + 6 i$ was found by solving for $x$ when $x - (1 + 6 i) = y$ and $y = 0$ .

The factor is $x - 1 - 6 i$

Step 4

Combine all the factors into a single equation.

$y = (x - 1 + 6 i) (x - 1 - 6 i)$

Step 5

Multiply all the factors to simplify the equation $y = (x - 1 + 6 i) (x - 1 - 6 i)$ .

Expand $(x - 1 + 6 i) (x - 1 - 6 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot - 1 + x (- 6 i) - 1 x - 1 \cdot - 1 - 1 (- 6 i) + 6 i x + 6 i \cdot - 1 + 6 i (- 6 i)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 1 + x (- 6 i) - 1 x - 1 \cdot - 1 - 1 (- 6 i) + 6 i x + 6 i \cdot - 1 + 6 i (- 6 i)$ .

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

$y = x \cdot x + x \cdot - 1 - 6 i x - 1 x - 1 \cdot - 1 - 1 (- 6 i) + 6 i x + 6 i \cdot - 1 + 6 i (- 6 i)$

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

$y = x \cdot x + x \cdot - 1 + 0 - 1 x - 1 \cdot - 1 - 1 (- 6 i) + 6 i \cdot - 1 + 6 i (- 6 i)$

Add $x \cdot x + x \cdot - 1$ and $0$ .

$y = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 (- 6 i) + 6 i \cdot - 1 + 6 i (- 6 i)$

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $- 6$ by $- 1$ .

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

Multiply $- 1$ by $6$ .

$y = x^{2} - x - x + 1 + 6 i - 6 i + 6 i (- 6 i)$

Multiply $6 i (- 6 i)$ .

Multiply $- 6$ by $6$ .

$y = x^{2} - x - x + 1 + 6 i - 6 i - 36 i i$

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

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

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

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

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

$y = x^{2} - x - x + 1 + 6 i - 6 i - 36 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{2} - x - x + 1 + 6 i - 6 i - 36 i^{2}$

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

$y = x^{2} - x - x + 1 + 6 i - 6 i - 36 \cdot - 1$

Multiply $- 36$ by $- 1$ .

$y = x^{2} - x - x + 1 + 6 i - 6 i + 36$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - x - x + 1 + 6 i - 6 i + 36$ .

Subtract $6 i$ from $6 i$ .

$y = x^{2} - x - x + 1 + 0 + 36$

Add $x^{2} - x - x + 1$ and $0$ .

$y = x^{2} - x - x + 1 + 36$

Subtract $x$ from $- x$ .

$y = x^{2} - 2 x + 1 + 36$

Add $1$ and $36$ .

$y = x^{2} - 2 x + 37$

Step 6

Find the Equation Given the Roots 1-6i

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