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

$2$ , $3$ , $\sqrt{5}$

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

The factor is $x - 2$

Step 3

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

The factor is $x - 3$

Step 4

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 5

Combine all the factors into a single equation.

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

Step 6

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

Expand $(x - 2) (x - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $- 2$ by $- 3$ .

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

Subtract $2 x$ from $- 3 x$ .

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

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

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

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 $1$ .

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

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

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

Add $2$ and $1$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $- 5$ .

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

Multiply $- 1$ by $6$ .

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

Reorder factors in $x^{3} + x^{2} (- \sqrt{5}) - 5 x^{2} + 5 x \sqrt{5} + 6 x - 6 \sqrt{5}$ .

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

Step 7

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

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