Find the Equation Given the Roots 3-5i , 3+5i

$3 - 5 i$ , $3 + 5 i$

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 $x = 3 - 5 i$ was found by solving for $x$ when $x - (3 - 5 i) = y$ and $y = 0$ .

The factor is $x - 3 + 5 i$

Step 3

The root at $x = 3 + 5 i$ was found by solving for $x$ when $x - (3 + 5 i) = y$ and $y = 0$ .

The factor is $x - 3 - 5 i$

Step 4

Combine all the factors into a single equation.

$y = (x - 3 + 5 i) (x - 3 - 5 i)$

Step 5

Multiply all the factors to simplify the equation $y = (x - 3 + 5 i) (x - 3 - 5 i)$ .

Expand $(x - 3 + 5 i) (x - 3 - 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot - 3 + x (- 5 i) - 3 x - 3 \cdot - 3 - 3 (- 5 i) + 5 i x + 5 i \cdot - 3 + 5 i (- 5 i)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 3 + x (- 5 i) - 3 x - 3 \cdot - 3 - 3 (- 5 i) + 5 i x + 5 i \cdot - 3 + 5 i (- 5 i)$ .

Reorder the factors in the terms $x (- 5 i)$ and $5 i x$ .

$y = x \cdot x + x \cdot - 3 - 5 i x - 3 x - 3 \cdot - 3 - 3 (- 5 i) + 5 i x + 5 i \cdot - 3 + 5 i (- 5 i)$

Add $- 5 i x$ and $5 i x$ .

$y = x \cdot x + x \cdot - 3 + 0 - 3 x - 3 \cdot - 3 - 3 (- 5 i) + 5 i \cdot - 3 + 5 i (- 5 i)$

Add $x \cdot x + x \cdot - 3$ and $0$ .

$y = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 (- 5 i) + 5 i \cdot - 3 + 5 i (- 5 i)$

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 (- 5 i) + 5 i \cdot - 3 + 5 i (- 5 i)$

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

$y = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 - 3 (- 5 i) + 5 i \cdot - 3 + 5 i (- 5 i)$

Multiply $- 3$ by $- 3$ .

$y = x^{2} - 3 x - 3 x + 9 - 3 (- 5 i) + 5 i \cdot - 3 + 5 i (- 5 i)$

Multiply $- 5$ by $- 3$ .

$y = x^{2} - 3 x - 3 x + 9 + 15 i + 5 i \cdot - 3 + 5 i (- 5 i)$

Multiply $- 3$ by $5$ .

$y = x^{2} - 3 x - 3 x + 9 + 15 i - 15 i + 5 i (- 5 i)$

Multiply $5 i (- 5 i)$ .

Multiply $- 5$ by $5$ .

$y = x^{2} - 3 x - 3 x + 9 + 15 i - 15 i - 25 i i$

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

$y = x^{2} - 3 x - 3 x + 9 + 15 i - 15 i - 25 (i i)$

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

$y = x^{2} - 3 x - 3 x + 9 + 15 i - 15 i - 25 (i i)$

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

$y = x^{2} - 3 x - 3 x + 9 + 15 i - 15 i - 25 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{2} - 3 x - 3 x + 9 + 15 i - 15 i - 25 i^{2}$

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

$y = x^{2} - 3 x - 3 x + 9 + 15 i - 15 i - 25 \cdot - 1$

Multiply $- 25$ by $- 1$ .

$y = x^{2} - 3 x - 3 x + 9 + 15 i - 15 i + 25$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 3 x - 3 x + 9 + 15 i - 15 i + 25$ .

Subtract $15 i$ from $15 i$ .

$y = x^{2} - 3 x - 3 x + 9 + 0 + 25$

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

$y = x^{2} - 3 x - 3 x + 9 + 25$

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

$y = x^{2} - 6 x + 9 + 25$

Add $9$ and $25$ .

$y = x^{2} - 6 x + 34$

Step 6

Find the Equation Given the Roots 3-5i , 3+5i

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