Find the Equation Given the Roots 0 , square root of 5 , 2

$0$ , $\sqrt{5}$ , $2$

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

The factor is $x$

Step 3

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

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

Step 4

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

The factor is $x - 2$

Step 5

Combine all the factors into a single equation.

$y = (x) (x - \sqrt{5}) (x - 2)$

Step 6

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

Simplify by multiplying through.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Expand $(x^{2} + x (- \sqrt{5})) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

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 $1$ .

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 1$ .

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

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

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

Reorder factors in $x^{3} - 2 x^{2} + x^{2} (- \sqrt{5}) + 2 x \sqrt{5}$ .

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

Step 7

Find the Equation Given the Roots 0 , square root of 5 , 2

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