Find the Equation Given the Roots 2+5i , 2-5i

$2 + 5 i$ , $2 - 5 i$

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

The root at $x = 2 + 5 i$ was found by solving for $x$ when $x - (2 + 5 i) = y$ and $y = 0$ .

The factor is $x - 2 - 5 i$

The root at $x = 2 - 5 i$ was found by solving for $x$ when $x - (2 - 5 i) = y$ and $y = 0$ .

The factor is $x - 2 + 5 i$

Combine all the factors into a single equation.

$y = (x - 2 - 5 i) (x - 2 + 5 i)$

Multiply all the factors to simplify the equation $y = (x - 2 - 5 i) (x - 2 + 5 i)$ .

Expand $(x - 2 - 5 i) (x - 2 + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot - 2 + x (5 i) - 2 x - 2 \cdot - 2 - 2 (5 i) - 5 i x - 5 i \cdot - 2 - 5 i (5 i)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 2 + x (5 i) - 2 x - 2 \cdot - 2 - 2 (5 i) - 5 i x - 5 i \cdot - 2 - 5 i (5 i)$ .

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

$y = x \cdot x + x \cdot - 2 + 5 i x - 2 x - 2 \cdot - 2 - 2 (5 i) - 5 i x - 5 i \cdot - 2 - 5 i (5 i)$

Subtract $5 i x$ from $5 i x$ .

$y = x \cdot x + x \cdot - 2 + 0 - 2 x - 2 \cdot - 2 - 2 (5 i) - 5 i \cdot - 2 - 5 i (5 i)$

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

$y = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 2 (5 i) - 5 i \cdot - 2 - 5 i (5 i)$

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 2$ by $- 2$ .

$y = x^{2} - 2 x - 2 x + 4 - 2 (5 i) - 5 i \cdot - 2 - 5 i (5 i)$

Multiply $5$ by $- 2$ .

$y = x^{2} - 2 x - 2 x + 4 - 10 i - 5 i \cdot - 2 - 5 i (5 i)$

Multiply $- 2$ by $- 5$ .

$y = x^{2} - 2 x - 2 x + 4 - 10 i + 10 i - 5 i (5 i)$

Multiply $- 5 i (5 i)$ .

Multiply $5$ by $- 5$ .

$y = x^{2} - 2 x - 2 x + 4 - 10 i + 10 i - 25 i i$

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

$y = x^{2} - 2 x - 2 x + 4 - 10 i + 10 i - 25 (i i)$

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

$y = x^{2} - 2 x - 2 x + 4 - 10 i + 10 i - 25 (i i)$

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

$y = x^{2} - 2 x - 2 x + 4 - 10 i + 10 i - 25 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{2} - 2 x - 2 x + 4 - 10 i + 10 i - 25 i^{2}$

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

$y = x^{2} - 2 x - 2 x + 4 - 10 i + 10 i - 25 \cdot - 1$

Multiply $- 25$ by $- 1$ .

$y = x^{2} - 2 x - 2 x + 4 - 10 i + 10 i + 25$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 2 x - 2 x + 4 - 10 i + 10 i + 25$ .

Add $- 10 i$ and $10 i$ .

$y = x^{2} - 2 x - 2 x + 4 + 0 + 25$

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

$y = x^{2} - 2 x - 2 x + 4 + 25$

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

$y = x^{2} - 4 x + 4 + 25$

Add $4$ and $25$ .

$y = x^{2} - 4 x + 29$

Find the Equation Given the Roots 2+5i , 2-5i

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