Find the Equation Given the Roots 2-7i

$2 - 7 i$

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 $x = 2 - 7 i$ was found by solving for $x$ when $x - (2 - 7 i) = y$ and $y = 0$ .

The factor is $x - 2 + 7 i$

Step 3

The root at $x = 2 + 7 i$ was found by solving for $x$ when $x - (2 + 7 i) = y$ and $y = 0$ .

The factor is $x - 2 - 7 i$

Step 4

Combine all the factors into a single equation.

$y = (x - 2 + 7 i) (x - 2 - 7 i)$

Step 5

Multiply all the factors to simplify the equation $y = (x - 2 + 7 i) (x - 2 - 7 i)$ .

Expand $(x - 2 + 7 i) (x - 2 - 7 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot - 2 + x (- 7 i) - 2 x - 2 \cdot - 2 - 2 (- 7 i) + 7 i x + 7 i \cdot - 2 + 7 i (- 7 i)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 2 + x (- 7 i) - 2 x - 2 \cdot - 2 - 2 (- 7 i) + 7 i x + 7 i \cdot - 2 + 7 i (- 7 i)$ .

Reorder the factors in the terms $x (- 7 i)$ and $7 i x$ .

$y = x \cdot x + x \cdot - 2 - 7 i x - 2 x - 2 \cdot - 2 - 2 (- 7 i) + 7 i x + 7 i \cdot - 2 + 7 i (- 7 i)$

Add $- 7 i x$ and $7 i x$ .

$y = x \cdot x + x \cdot - 2 + 0 - 2 x - 2 \cdot - 2 - 2 (- 7 i) + 7 i \cdot - 2 + 7 i (- 7 i)$

Add $x \cdot x + x \cdot - 2$ and $0$ .

$y = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 2 (- 7 i) + 7 i \cdot - 2 + 7 i (- 7 i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - 2 (- 7 i) + 7 i \cdot - 2 + 7 i (- 7 i)$

Move $- 2$ to the left of $x$ .

$y = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - 2 (- 7 i) + 7 i \cdot - 2 + 7 i (- 7 i)$

Multiply $- 2$ by $- 2$ .

$y = x^{2} - 2 x - 2 x + 4 - 2 (- 7 i) + 7 i \cdot - 2 + 7 i (- 7 i)$

Multiply $- 7$ by $- 2$ .

$y = x^{2} - 2 x - 2 x + 4 + 14 i + 7 i \cdot - 2 + 7 i (- 7 i)$

Multiply $- 2$ by $7$ .

$y = x^{2} - 2 x - 2 x + 4 + 14 i - 14 i + 7 i (- 7 i)$

Multiply $7 i (- 7 i)$ .

Multiply $- 7$ by $7$ .

$y = x^{2} - 2 x - 2 x + 4 + 14 i - 14 i - 49 i i$

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

$y = x^{2} - 2 x - 2 x + 4 + 14 i - 14 i - 49 (i i)$

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

$y = x^{2} - 2 x - 2 x + 4 + 14 i - 14 i - 49 (i i)$

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

$y = x^{2} - 2 x - 2 x + 4 + 14 i - 14 i - 49 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{2} - 2 x - 2 x + 4 + 14 i - 14 i - 49 i^{2}$

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

$y = x^{2} - 2 x - 2 x + 4 + 14 i - 14 i - 49 \cdot - 1$

Multiply $- 49$ by $- 1$ .

$y = x^{2} - 2 x - 2 x + 4 + 14 i - 14 i + 49$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 2 x - 2 x + 4 + 14 i - 14 i + 49$ .

Subtract $14 i$ from $14 i$ .

$y = x^{2} - 2 x - 2 x + 4 + 0 + 49$

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

$y = x^{2} - 2 x - 2 x + 4 + 49$

Subtract $2 x$ from $- 2 x$ .

$y = x^{2} - 4 x + 4 + 49$

Add $4$ and $49$ .

$y = x^{2} - 4 x + 53$

Step 6

Find the Equation Given the Roots 2-7i

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