Find the Equation Given the Roots 4+3i , 2+ square root of 5

$4 + 3 i$ , $2 + \sqrt{5}$

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 $x = 4 + 3 i$ was found by solving for $x$ when $x - (4 + 3 i) = y$ and $y = 0$ .

The factor is $x - 4 - 3 i$

Step 3

The root at $x = 4 - 3 i$ was found by solving for $x$ when $x - (4 - 3 i) = y$ and $y = 0$ .

The factor is $x - 4 + 3 i$

Step 4

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

The factor is $x - 2 - \sqrt{5}$

Step 5

Combine all the factors into a single equation.

$y = (x - 4 - 3 i) (x - 4 + 3 i) (x - 2 - \sqrt{5})$

Step 6

Multiply all the factors to simplify the equation $y = (x - 4 - 3 i) (x - 4 + 3 i) (x - 2 - \sqrt{5})$ .

Expand $(x - 4 - 3 i) (x - 4 + 3 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 4 + x (3 i) - 4 x - 4 \cdot - 4 - 4 (3 i) - 3 i x - 3 i \cdot - 4 - 3 i (3 i)) (x - 2 - \sqrt{5})$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 4 + x (3 i) - 4 x - 4 \cdot - 4 - 4 (3 i) - 3 i x - 3 i \cdot - 4 - 3 i (3 i)$ .

Reorder the factors in the terms $x (3 i)$ and $- 3 i x$ .

$y = (x \cdot x + x \cdot - 4 + 3 i x - 4 x - 4 \cdot - 4 - 4 (3 i) - 3 i x - 3 i \cdot - 4 - 3 i (3 i)) (x - 2 - \sqrt{5})$

Subtract $3 i x$ from $3 i x$ .

$y = (x \cdot x + x \cdot - 4 + 0 - 4 x - 4 \cdot - 4 - 4 (3 i) - 3 i \cdot - 4 - 3 i (3 i)) (x - 2 - \sqrt{5})$

Add $x \cdot x + x \cdot - 4$ and $0$ .

$y = (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 - 4 (3 i) - 3 i \cdot - 4 - 3 i (3 i)) (x - 2 - \sqrt{5})$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4 - 4 (3 i) - 3 i \cdot - 4 - 3 i (3 i)) (x - 2 - \sqrt{5})$

Move $- 4$ to the left of $x$ .

$y = (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4 - 4 (3 i) - 3 i \cdot - 4 - 3 i (3 i)) (x - 2 - \sqrt{5})$

Multiply $- 4$ by $- 4$ .

$y = (x^{2} - 4 x - 4 x + 16 - 4 (3 i) - 3 i \cdot - 4 - 3 i (3 i)) (x - 2 - \sqrt{5})$

Multiply $3$ by $- 4$ .

$y = (x^{2} - 4 x - 4 x + 16 - 12 i - 3 i \cdot - 4 - 3 i (3 i)) (x - 2 - \sqrt{5})$

Multiply $- 4$ by $- 3$ .

$y = (x^{2} - 4 x - 4 x + 16 - 12 i + 12 i - 3 i (3 i)) (x - 2 - \sqrt{5})$

Multiply $- 3 i (3 i)$ .

Multiply $3$ by $- 3$ .

$y = (x^{2} - 4 x - 4 x + 16 - 12 i + 12 i - 9 i i) (x - 2 - \sqrt{5})$

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

$y = (x^{2} - 4 x - 4 x + 16 - 12 i + 12 i - 9 (i i)) (x - 2 - \sqrt{5})$

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

$y = (x^{2} - 4 x - 4 x + 16 - 12 i + 12 i - 9 (i i)) (x - 2 - \sqrt{5})$

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

$y = (x^{2} - 4 x - 4 x + 16 - 12 i + 12 i - 9 i^{1 + 1}) (x - 2 - \sqrt{5})$

Add $1$ and $1$ .

$y = (x^{2} - 4 x - 4 x + 16 - 12 i + 12 i - 9 i^{2}) (x - 2 - \sqrt{5})$

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

$y = (x^{2} - 4 x - 4 x + 16 - 12 i + 12 i - 9 \cdot - 1) (x - 2 - \sqrt{5})$

Multiply $- 9$ by $- 1$ .

$y = (x^{2} - 4 x - 4 x + 16 - 12 i + 12 i + 9) (x - 2 - \sqrt{5})$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 4 x - 4 x + 16 - 12 i + 12 i + 9$ .

Add $- 12 i$ and $12 i$ .

$y = (x^{2} - 4 x - 4 x + 16 + 0 + 9) (x - 2 - \sqrt{5})$

Add $x^{2} - 4 x - 4 x + 16$ and $0$ .

$y = (x^{2} - 4 x - 4 x + 16 + 9) (x - 2 - \sqrt{5})$

Subtract $4 x$ from $- 4 x$ .

$y = (x^{2} - 8 x + 16 + 9) (x - 2 - \sqrt{5})$

Add $16$ and $9$ .

$y = (x^{2} - 8 x + 25) (x - 2 - \sqrt{5})$

Expand $(x^{2} - 8 x + 25) (x - 2 - \sqrt{5})$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 2 + x^{2} (- \sqrt{5}) - 8 x \cdot x - 8 x \cdot - 2 - 8 x (- \sqrt{5}) + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

Simplify terms.

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

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

$y = x^{2} x + x^{2} \cdot - 2 + x^{2} (- \sqrt{5}) - 8 x \cdot x - 8 x \cdot - 2 - 8 x (- \sqrt{5}) + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

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

$y = x^{2 + 1} + x^{2} \cdot - 2 + x^{2} (- \sqrt{5}) - 8 x \cdot x - 8 x \cdot - 2 - 8 x (- \sqrt{5}) + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 2 + x^{2} (- \sqrt{5}) - 8 x \cdot x - 8 x \cdot - 2 - 8 x (- \sqrt{5}) + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

Move $- 2$ to the left of $x^{2}$ .

$y = x^{3} - 2 \cdot x^{2} + x^{2} (- \sqrt{5}) - 8 x \cdot x - 8 x \cdot - 2 - 8 x (- \sqrt{5}) + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y = x^{3} - 2 x^{2} + x^{2} (- \sqrt{5}) - 8 (x \cdot x) - 8 x \cdot - 2 - 8 x (- \sqrt{5}) + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

Multiply $x$ by $x$ .

$y = x^{3} - 2 x^{2} + x^{2} (- \sqrt{5}) - 8 x^{2} - 8 x \cdot - 2 - 8 x (- \sqrt{5}) + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

Multiply $- 2$ by $- 8$ .

$y = x^{3} - 2 x^{2} + x^{2} (- \sqrt{5}) - 8 x^{2} + 16 x - 8 x (- \sqrt{5}) + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

Multiply $- 1$ by $- 8$ .

$y = x^{3} - 2 x^{2} + x^{2} (- \sqrt{5}) - 8 x^{2} + 16 x + 8 x \sqrt{5} + 25 x + 25 \cdot - 2 + 25 (- \sqrt{5})$

Multiply $25$ by $- 2$ .

$y = x^{3} - 2 x^{2} + x^{2} (- \sqrt{5}) - 8 x^{2} + 16 x + 8 x \sqrt{5} + 25 x - 50 + 25 (- \sqrt{5})$

Multiply $- 1$ by $25$ .

$y = x^{3} - 2 x^{2} + x^{2} (- \sqrt{5}) - 8 x^{2} + 16 x + 8 x \sqrt{5} + 25 x - 50 - 25 \sqrt{5}$

Simplify by adding terms.

Subtract $8 x^{2}$ from $- 2 x^{2}$ .

$y = x^{3} - 10 x^{2} + x^{2} (- \sqrt{5}) + 16 x + 8 x \sqrt{5} + 25 x - 50 - 25 \sqrt{5}$

Add $16 x$ and $25 x$ .

$y = x^{3} - 10 x^{2} + x^{2} (- \sqrt{5}) + 8 x \sqrt{5} + 41 x - 50 - 25 \sqrt{5}$

Reorder factors in $x^{3} - 10 x^{2} + x^{2} (- \sqrt{5}) + 8 x \sqrt{5} + 41 x - 50 - 25 \sqrt{5}$ .

$y = x^{3} - 10 x^{2} - x^{2} \sqrt{5} + 8 x \sqrt{5} + 41 x - 50 - 25 \sqrt{5}$

Step 7

Find the Equation Given the Roots 4+3i , 2+ square root of 5

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