Find the Equation Given the Roots -6 , 0 , 3- square root of 5

$- 6$ , $0$ , $3 - \sqrt{5}$

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 $x = - 6$ was found by solving for $x$ when $x - (- 6) = y$ and $y = 0$ .

The factor is $x + 6$

Step 3

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

The factor is $x$

Step 4

The root at $x = 3 - \sqrt{5}$ was found by solving for $x$ when $x - (3 - \sqrt{5}) = y$ and $y = 0$ .

The factor is $x - 3 + \sqrt{5}$

Step 5

Combine all the factors into a single equation.

$y = (x + 6) (x) (x - 3 + \sqrt{5})$

Step 6

Multiply all the factors to simplify the equation $y = (x + 6) (x) (x - 3 + \sqrt{5})$ .

Simplify by multiplying through.

Apply the distributive property.

$y = (x \cdot x + 6 x) (x - 3 + \sqrt{5})$

Multiply $x$ by $x$ .

$y = (x^{2} + 6 x) (x - 3 + \sqrt{5})$

Expand $(x^{2} + 6 x) (x - 3 + \sqrt{5})$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 3 + x^{2} \sqrt{5} + 6 x \cdot x + 6 x \cdot - 3 + 6 x \sqrt{5}$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 3 + x^{2} \sqrt{5} + 6 x \cdot x + 6 x \cdot - 3 + 6 x \sqrt{5}$

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

$y = x^{2 + 1} + x^{2} \cdot - 3 + x^{2} \sqrt{5} + 6 x \cdot x + 6 x \cdot - 3 + 6 x \sqrt{5}$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 3 + x^{2} \sqrt{5} + 6 x \cdot x + 6 x \cdot - 3 + 6 x \sqrt{5}$

Move $- 3$ to the left of $x^{2}$ .

$y = x^{3} - 3 \cdot x^{2} + x^{2} \sqrt{5} + 6 x \cdot x + 6 x \cdot - 3 + 6 x \sqrt{5}$

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

Move $x$ .

$y = x^{3} - 3 x^{2} + x^{2} \sqrt{5} + 6 (x \cdot x) + 6 x \cdot - 3 + 6 x \sqrt{5}$

Multiply $x$ by $x$ .

$y = x^{3} - 3 x^{2} + x^{2} \sqrt{5} + 6 x^{2} + 6 x \cdot - 3 + 6 x \sqrt{5}$

Multiply $- 3$ by $6$ .

$y = x^{3} - 3 x^{2} + x^{2} \sqrt{5} + 6 x^{2} - 18 x + 6 x \sqrt{5}$

Add $- 3 x^{2}$ and $6 x^{2}$ .

$y = x^{3} + 3 x^{2} + x^{2} \sqrt{5} - 18 x + 6 x \sqrt{5}$

Step 7

Find the Equation Given the Roots -6 , 0 , 3- square root of 5

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