Algebra Examples

$(- \frac{\sqrt{3}}{2}, \frac{1}{2})$

$x = - \frac{\sqrt{3}}{2}$ and $x = \frac{1}{2}$ are the two real distinct solutions for the quadratic equation, which means that $x + \frac{\sqrt{3}}{2}$ and $x - \frac{1}{2}$ are the factors of the quadratic equation.

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

Expand $(x + \frac{\sqrt{3}}{2}) (x - \frac{1}{2})$ using the FOIL Method.

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Apply the distributive property.

$x (x - \frac{1}{2}) + \frac{\sqrt{3}}{2} \cdot (x - \frac{1}{2}) = 0$

Apply the distributive property.

$x \cdot x + x (- \frac{1}{2}) + \frac{\sqrt{3}}{2} \cdot (x - \frac{1}{2}) = 0$

Apply the distributive property.

$x \cdot x + x (- \frac{1}{2}) + \frac{\sqrt{3}}{2} x + \frac{\sqrt{3}}{2} \cdot (- \frac{1}{2}) = 0$

Simplify each term.

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Multiply $x$ by $x$ .

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

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

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

Combine $\frac{\sqrt{3}}{2}$ and $x$ .

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

Multiply $\frac{\sqrt{3}}{2} (- \frac{1}{2})$ .

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Multiply $\frac{\sqrt{3}}{2}$ by $\frac{1}{2}$ .

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

Multiply $2$ by $2$ .

$x^{2} - \frac{x}{2} + \frac{\sqrt{3} x}{2} - \frac{\sqrt{3}}{4} = 0$

The standard quadratic equation using the given set of solutions ${- \frac{\sqrt{3}}{2}, \frac{1}{2}}$ is $y = x^{2} - \frac{x}{2} + \frac{\sqrt{3} x}{2} - \frac{\sqrt{3}}{4}$ .

$y = x^{2} - \frac{x}{2} + \frac{\sqrt{3} x}{2} - \frac{\sqrt{3}}{4}$