Finite Math Examples

$x = \sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{x}}$

Step 1

Simplify $\sqrt{19} + \frac{91}{\sqrt{19} + \frac{91}{x}}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify the denominator.

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Step 1.1.1.1

To write $\sqrt{19}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$x = \sqrt{19} + \frac{91}{\frac{\sqrt{19} x}{x} + \frac{91}{x}}$

Step 1.1.1.2

Combine the numerators over the common denominator.

$x = \sqrt{19} + \frac{91}{\frac{\sqrt{19} x + 91}{x}}$

Step 1.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = \sqrt{19} + 91 \frac{x}{\sqrt{19} x + 91}$

Step 1.1.3

Multiply $\frac{x}{\sqrt{19} x + 91}$ by $\frac{\sqrt{19} x - 91}{\sqrt{19} x - 91}$ .

$x = \sqrt{19} + 91 (\frac{x}{\sqrt{19} x + 91} \cdot \frac{\sqrt{19} x - 91}{\sqrt{19} x - 91})$

Step 1.1.4

Multiply $\frac{x}{\sqrt{19} x + 91}$ by $\frac{\sqrt{19} x - 91}{\sqrt{19} x - 91}$ .

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{(\sqrt{19} x + 91) (\sqrt{19} x - 91)}$

Step 1.1.5

Expand the denominator using the FOIL method.

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{x^{2} {\sqrt{19}}^{2} + \sqrt{19} x \cdot - 91 + 91 (\sqrt{19} x) - 8281}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 1.1.6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{x^{2} {(19^{\frac{1}{2}})}^{2} - 8281}$

Step 1.1.6.1.2

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

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{x^{2} \cdot 19^{\frac{1}{2} \cdot 2} - 8281}$

Step 1.1.6.1.3

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

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{x^{2} \cdot 19^{\frac{2}{2}} - 8281}$

Step 1.1.6.1.4

Cancel the common factor of $2$ .

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Step 1.1.6.1.4.1

Cancel the common factor.

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{x^{2} \cdot 19^{\frac{2}{2}} - 8281}$

Step 1.1.6.1.4.2

Rewrite the expression.

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{x^{2} \cdot 19^{1} - 8281}$

Step 1.1.6.1.5

Evaluate the exponent.

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{x^{2} \cdot 19 - 8281}$

Step 1.1.6.2

Move $19$ to the left of $x^{2}$ .

$x = \sqrt{19} + 91 \frac{x (\sqrt{19} x - 91)}{19 x^{2} - 8281}$

Step 1.1.7

Combine $91$ and $\frac{x (\sqrt{19} x - 91)}{19 x^{2} - 8281}$ .

$x = \sqrt{19} + \frac{91 x (\sqrt{19} x - 91)}{19 x^{2} - 8281}$

Step 1.2

To write $\sqrt{19}$ as a fraction with a common denominator, multiply by $\frac{19 x^{2} - 8281}{19 x^{2} - 8281}$ .

$x = \frac{\sqrt{19} (19 x^{2} - 8281)}{19 x^{2} - 8281} + \frac{91 x (\sqrt{19} x - 91)}{19 x^{2} - 8281}$

Step 1.3

Combine the numerators over the common denominator.

$x = \frac{\sqrt{19} (19 x^{2} - 8281) + 91 x (\sqrt{19} x - 91)}{19 x^{2} - 8281}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$x = \frac{\sqrt{19} (19 x^{2}) + \sqrt{19} \cdot - 8281 + 91 x (\sqrt{19} x - 91)}{19 x^{2} - 8281}$

Step 1.4.2

Move $- 8281$ to the left of $\sqrt{19}$ .

$x = \frac{\sqrt{19} \cdot 19 x^{2} - 8281 \cdot \sqrt{19} + 91 x (\sqrt{19} x - 91)}{19 x^{2} - 8281}$

Step 1.4.3

Move $19$ to the left of $\sqrt{19}$ .

$x = \frac{19 \sqrt{19} x^{2} - 8281 \sqrt{19} + 91 x (\sqrt{19} x - 91)}{19 x^{2} - 8281}$

Step 1.4.4

Apply the distributive property.

$x = \frac{19 \sqrt{19} x^{2} - 8281 \sqrt{19} + 91 x (\sqrt{19} x) + 91 x \cdot - 91}{19 x^{2} - 8281}$

Step 1.4.5

Rewrite using the commutative property of multiplication.

$x = \frac{19 \sqrt{19} x^{2} - 8281 \sqrt{19} + 91 \cdot \sqrt{19} x \cdot x + 91 x \cdot - 91}{19 x^{2} - 8281}$

Step 1.4.6

Multiply $- 91$ by $91$ .

$x = \frac{19 \sqrt{19} x^{2} - 8281 \sqrt{19} + 91 \cdot \sqrt{19} x \cdot x - 8281 x}{19 x^{2} - 8281}$

Step 1.4.7

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.7.1

Move $x$ .

$x = \frac{19 \sqrt{19} x^{2} - 8281 \sqrt{19} + 91 \cdot \sqrt{19} (x \cdot x) - 8281 x}{19 x^{2} - 8281}$

Step 1.4.7.2

Multiply $x$ by $x$ .

$x = \frac{19 \sqrt{19} x^{2} - 8281 \sqrt{19} + 91 \cdot \sqrt{19} x^{2} - 8281 x}{19 x^{2} - 8281}$

$x = \frac{19 \sqrt{19} x^{2} - 8281 \sqrt{19} + 91 \sqrt{19} x^{2} - 8281 x}{19 x^{2} - 8281}$

Step 1.4.8

Add $19 \sqrt{19} x^{2}$ and $91 \sqrt{19} x^{2}$ .

$x = \frac{110 \sqrt{19} x^{2} - 8281 \sqrt{19} - 8281 x}{19 x^{2} - 8281}$

Step 1.4.9

Reorder terms.

$x = \frac{110 x^{2} \sqrt{19} - 8281 x - 8281 \sqrt{19}}{19 x^{2} - 8281}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 7.60574342, 11.96464236$

Step 3