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

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

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

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

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.

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

Step 6

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify terms.

Combine the opposite terms in $x \cdot x + x (- \sqrt{3}) + \sqrt{3} x + \sqrt{3} (- \sqrt{3})$ .

Reorder the factors in the terms $x (- \sqrt{3})$ and $\sqrt{3} x$ .

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

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

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

Add $x \cdot x$ and $0$ .

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $\sqrt{3} (- \sqrt{3})$ .

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = (x^{2} - {(3^{\frac{1}{2}})}^{2}) (x + 2)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $3$ .

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

Expand $(x^{2} - 3) (x + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

Multiply $- 3$ by $2$ .

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

Step 7

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

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