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

$\sqrt{3}$ , $5 i$

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 $x = \sqrt{3}$ was found by solving for $x$ when $x - (\sqrt{3}) = y$ and $y = 0$ .

The factor is $x - \sqrt{3}$

Step 3

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

The factor is $x - 5 i$

Step 4

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

The factor is $x + 5 i$

Step 5

Combine all the factors into a single equation.

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

Step 6

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

Expand $(x - \sqrt{3}) (x - 5 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- 5$ by $- 1$ .

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

Expand $(x^{2} + x (- 5 i) - \sqrt{3} x + 5 \sqrt{3} i) (x + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

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

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 $1$ .

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

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

$y = x^{2 + 1} + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) - \sqrt{3} x \cdot x - \sqrt{3} x (5 i) + 5 \sqrt{3} i x + 5 \sqrt{3} i (5 i)$

Add $2$ and $1$ .

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

Move $5$ to the left of $x^{2}$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $x (- 5 i) (5 i)$ .

Multiply $5$ by $- 5$ .

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

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

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

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

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

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 i^{1 + 1}) - \sqrt{3} x \cdot x - \sqrt{3} x (5 i) + 5 \sqrt{3} i x + 5 \sqrt{3} i (5 i)$

Add $1$ and $1$ .

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

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 \cdot - 1) - \sqrt{3} x \cdot x - \sqrt{3} x (5 i) + 5 \sqrt{3} i x + 5 \sqrt{3} i (5 i)$

Multiply $- 25$ by $- 1$ .

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

Move $25$ to the left of $x$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $5$ by $- 1$ .

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

Multiply $5 \sqrt{3} i (5 i)$ .

Multiply $5$ by $5$ .

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

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

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

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

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

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - \sqrt{3} x^{2} - 5 \sqrt{3} x i + 5 \sqrt{3} i x + 25 \sqrt{3} i^{1 + 1}$

Add $1$ and $1$ .

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

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - \sqrt{3} x^{2} - 5 \sqrt{3} x i + 5 \sqrt{3} i x + 25 \sqrt{3} \cdot - 1$

Multiply $- 1$ by $25$ .

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

Simplify by adding terms.

Combine the opposite terms in $x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x - \sqrt{3} x^{2} - 5 \sqrt{3} x i + 5 \sqrt{3} i x - 25 \sqrt{3}$ .

Reorder the factors in the terms $5 x^{2} i$ and $x^{2} (- 5 i)$ .

$y = x^{3} + 5 i x^{2} - 5 i x^{2} + 25 x - \sqrt{3} x^{2} - 5 \sqrt{3} x i + 5 \sqrt{3} i x - 25 \sqrt{3}$

Subtract $5 i x^{2}$ from $5 i x^{2}$ .

$y = x^{3} + 0 + 25 x - \sqrt{3} x^{2} - 5 \sqrt{3} x i + 5 \sqrt{3} i x - 25 \sqrt{3}$

Add $x^{3}$ and $0$ .

$y = x^{3} + 25 x - \sqrt{3} x^{2} - 5 \sqrt{3} x i + 5 \sqrt{3} i x - 25 \sqrt{3}$

Reorder the factors in the terms $- 5 \sqrt{3} x i$ and $5 \sqrt{3} i x$ .

$y = x^{3} + 25 x - \sqrt{3} x^{2} - 5 i x \sqrt{3} + 5 i x \sqrt{3} - 25 \sqrt{3}$

Add $- 5 i x \sqrt{3}$ and $5 i x \sqrt{3}$ .

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

Add $x^{3} + 25 x - \sqrt{3} x^{2}$ and $0$ .

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

Move $25 x$ .

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

Step 7

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

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