Algebra Examples

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

Step 1

$x = - \sqrt{3}$ and $x = \sqrt{3}$ are the two real distinct solutions for the quadratic equation, which means that $x + \sqrt{3}$ and $x - \sqrt{3}$ are the factors of the quadratic equation.

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

Step 2

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify terms.

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

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

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

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

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

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

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

Simplify each term.

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Multiply $x$ by $x$ .

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

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

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

$x^{2} - (\sqrt{3} \sqrt{3}) = 0$

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

$x^{2} - (\sqrt{3} \sqrt{3}) = 0$

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

$x^{2} - {\sqrt{3}}^{1 + 1} = 0$

Add $1$ and $1$ .

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

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

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

$x^{2} - {(3^{\frac{1}{2}})}^{2} = 0$

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

$x^{2} - 3^{\frac{1}{2} \cdot 2} = 0$

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

$x^{2} - 3^{\frac{2}{2}} = 0$

Cancel the common factor of $2$ .

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Cancel the common factor.

$x^{2} - 3^{\frac{2}{2}} = 0$

Rewrite the expression.

$x^{2} - 3 = 0$

Evaluate the exponent.

$x^{2} - 1 \cdot 3 = 0$

Multiply $- 1$ by $3$ .

$x^{2} - 3 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${- \sqrt{3}, \sqrt{3}}$ is $y = x^{2} - 3$ .

$y = x^{2} - 3$

Step 5