Finite Math Examples

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

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

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

Step 3.1.2

Multiply $- 4$ by $- 4$ .

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

Step 3.1.3

Multiply $16 \sqrt{3} \sqrt{3}$ .

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Step 3.1.3.1

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

$x = \frac{- 1 \pm \sqrt{1 + 16 (\sqrt{3} \sqrt{3})}}{2 \sqrt{3}}$

Step 3.1.3.2

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

$x = \frac{- 1 \pm \sqrt{1 + 16 (\sqrt{3} \sqrt{3})}}{2 \sqrt{3}}$

Step 3.1.3.3

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

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

Step 3.1.3.4

Add $1$ and $1$ .

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

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 $3^{\frac{1}{2}}$ .

$x = \frac{- 1 \pm \sqrt{1 + 16 {(3^{\frac{1}{2}})}^{2}}}{2 \sqrt{3}}$

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

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.

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

Step 3.1.4.4.2

Rewrite the expression.

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

Step 3.1.4.5

Evaluate the exponent.

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

Step 3.1.5

Multiply $16$ by $3$ .

$x = \frac{- 1 \pm \sqrt{1 + 48}}{2 \sqrt{3}}$

Step 3.1.6

Add $1$ and $48$ .

$x = \frac{- 1 \pm \sqrt{49}}{2 \sqrt{3}}$

Step 3.1.7

Rewrite $49$ as $7^{2}$ .

$x = \frac{- 1 \pm \sqrt{7^{2}}}{2 \sqrt{3}}$

Step 3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 1 \pm 7}{2 \sqrt{3}}$

Step 3.2

Multiply $\frac{- 1 \pm 7}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{- 1 \pm 7}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{- 1 \pm 7}{2 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 \sqrt{3} \sqrt{3}}$

Step 3.3.2

Move $\sqrt{3}$ .

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 3.3.3

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

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 3.3.4

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

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 (\sqrt{3} \sqrt{3})}$

Step 3.3.5

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

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}}$

Step 3.3.6

Add $1$ and $1$ .

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 {\sqrt{3}}^{2}}$

Step 3.3.7

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

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Step 3.3.7.1

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

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}}$

Step 3.3.7.2

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

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 3.3.7.3

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

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 3.3.7.4

Cancel the common factor of $2$ .

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Step 3.3.7.4.1

Cancel the common factor.

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Step 3.3.7.4.2

Rewrite the expression.

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 \cdot 3}$

Step 3.3.7.5

Evaluate the exponent.

$x = \frac{(- 1 \pm 7) \sqrt{3}}{2 \cdot 3}$

Step 3.4

Multiply $2$ by $3$ .

$x = \frac{(- 1 \pm 7) \sqrt{3}}{6}$

Step 3.5

Factor $- 1$ out of $- 1 \pm 7$ .

$x = \frac{- 1 (- (- 1 \pm 7)) \sqrt{3}}{6}$

Step 3.6

Multiply $- 1$ by $- 1$ .

$x = \frac{1 (- 1 \pm 7) \sqrt{3}}{6}$

Step 3.7

Multiply $- 1 \pm 7$ by $1$ .

$x = \frac{(- 1 \pm 7) \sqrt{3}}{6}$

Step 3.8

Reorder factors in $\frac{(- 1 \pm 7) \sqrt{3}}{6}$ .

$x = \frac{\sqrt{3} (- 1 \pm 7)}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{3} (- 1 \pm 7)}{6}$

Decimal Form:

$x = 1.73205080 \dots, - 2.30940107 \dots$