Precalculus Examples

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

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 1$ , $b = - \sqrt{3}$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{\sqrt{3} \pm \sqrt{{(- \sqrt{3})}^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Apply the product rule to $- \sqrt{3}$ .

$x = \frac{\sqrt{3} \pm \sqrt{{(- 1)}^{2} {\sqrt{3}}^{2} - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{\sqrt{3} \pm \sqrt{1 {\sqrt{3}}^{2} - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.3

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

$x = \frac{\sqrt{3} \pm \sqrt{{\sqrt{3}}^{2} - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.4

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

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Step 3.1.4.1

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

$x = \frac{\sqrt{3} \pm \sqrt{{(3^{\frac{1}{2}})}^{2} - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.4.2

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

$x = \frac{\sqrt{3} \pm \sqrt{3^{\frac{1}{2} \cdot 2} - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.4.3

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

$x = \frac{\sqrt{3} \pm \sqrt{3^{\frac{2}{2}} - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.4.4.1

Cancel the common factor.

$x = \frac{\sqrt{3} \pm \sqrt{3^{\frac{2}{2}} - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.4.4.2

Rewrite the expression.

$x = \frac{\sqrt{3} \pm \sqrt{3 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.4.5

Evaluate the exponent.

$x = \frac{\sqrt{3} \pm \sqrt{3 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 3.1.5

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 3.1.5.1

Multiply $- 4$ by $1$ .

$x = \frac{\sqrt{3} \pm \sqrt{3 - 4 \cdot - 4}}{2 \cdot 1}$

Step 3.1.5.2

Multiply $- 4$ by $- 4$ .

$x = \frac{\sqrt{3} \pm \sqrt{3 + 16}}{2 \cdot 1}$

Step 3.1.6

Add $3$ and $16$ .

$x = \frac{\sqrt{3} \pm \sqrt{19}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{\sqrt{3} \pm \sqrt{19}}{2}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{\sqrt{3} + \sqrt{19}}{2}, \frac{\sqrt{3} - \sqrt{19}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{3} + \sqrt{19}}{2}, \frac{\sqrt{3} - \sqrt{19}}{2}$

Decimal Form:

$x = 3.04547487 \dots, - 1.31342406 \dots$