Solve 6x+6/4x-6+8+x*-2)=0

$$\frac{ 6x+6 }{ 4x } -6+8+x \times -2)=0$$

Evaluate

$$-2x+\frac{7}{2}+\frac{3}{2x}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{6x+6}{4x}.

$$\frac{6\left(x+1\right)}{4x}-6+8+x\left(-2\right)$$

Cancel out 2 in both numerator and denominator.

$$\frac{3\left(x+1\right)}{2x}-6+8+x\left(-2\right)$$

Add -6 and 8 to get 2.

$$\frac{3\left(x+1\right)}{2x}+2+x\left(-2\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2+x\left(-2\right) times \frac{2x}{2x}.

$$\frac{3\left(x+1\right)}{2x}+\frac{\left(2+x\left(-2\right)\right)\times 2x}{2x}$$

Since \frac{3\left(x+1\right)}{2x} and \frac{\left(2+x\left(-2\right)\right)\times 2x}{2x} have the same denominator, add them by adding their numerators.

$$\frac{3\left(x+1\right)+\left(2+x\left(-2\right)\right)\times 2x}{2x}$$

Do the multiplications in 3\left(x+1\right)+\left(2+x\left(-2\right)\right)\times 2x.

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

Combine like terms in 3x+3+4x-4x^{2}.

$$\frac{7x+3-4x^{2}}{2x}$$

Factor the expressions that are not already factored in \frac{7x+3-4x^{2}}{2x}.

$$\frac{-4\left(x-\left(-\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)}{2x}$$

Cancel out 2 in both numerator and denominator.

$$\frac{-2\left(x-\left(-\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)}{x}$$

To find the opposite of -\frac{1}{8}\sqrt{97}+\frac{7}{8}, find the opposite of each term.

$$\frac{-2\left(x-\left(-\frac{1}{8}\sqrt{97}\right)-\frac{7}{8}\right)\left(x-\left(\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)}{x}$$

The opposite of -\frac{1}{8}\sqrt{97} is \frac{1}{8}\sqrt{97}.

$$\frac{-2\left(x+\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)\left(x-\left(\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)}{x}$$

To find the opposite of \frac{1}{8}\sqrt{97}+\frac{7}{8}, find the opposite of each term.

$$\frac{-2\left(x+\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Use the distributive property to multiply -2 by x+\frac{1}{8}\sqrt{97}-\frac{7}{8}.

$$\frac{\left(-2x-2\times \frac{1}{8}\sqrt{97}-2\left(-\frac{7}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Multiply -2 and \frac{1}{8} to get \frac{-2}{8}.

$$\frac{\left(-2x+\frac{-2}{8}\sqrt{97}-2\left(-\frac{7}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Reduce the fraction \frac{-2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{\left(-2x-\frac{1}{4}\sqrt{97}-2\left(-\frac{7}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Express -2\left(-\frac{7}{8}\right) as a single fraction.

$$\frac{\left(-2x-\frac{1}{4}\sqrt{97}+\frac{-2\left(-7\right)}{8}\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Multiply -2 and -7 to get 14.

$$\frac{\left(-2x-\frac{1}{4}\sqrt{97}+\frac{14}{8}\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Reduce the fraction \frac{14}{8} to lowest terms by extracting and canceling out 2.

$$\frac{\left(-2x-\frac{1}{4}\sqrt{97}+\frac{7}{4}\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Apply the distributive property by multiplying each term of -2x-\frac{1}{4}\sqrt{97}+\frac{7}{4} by each term of x-\frac{1}{8}\sqrt{97}-\frac{7}{8}.

$$\frac{-2x^{2}-2x\left(-\frac{1}{8}\right)\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\sqrt{97}\left(-\frac{1}{8}\right)\sqrt{97}-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply \sqrt{97} and \sqrt{97} to get 97.

$$\frac{-2x^{2}-2x\left(-\frac{1}{8}\right)\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Express -2\left(-\frac{1}{8}\right) as a single fraction.

$$\frac{-2x^{2}+\frac{-2\left(-1\right)}{8}x\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply -2 and -1 to get 2.

$$\frac{-2x^{2}+\frac{2}{8}x\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{-2x^{2}+\frac{1}{4}x\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Express -2\left(-\frac{7}{8}\right) as a single fraction.

$$\frac{-2x^{2}+\frac{1}{4}x\sqrt{97}+\frac{-2\left(-7\right)}{8}x-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply -2 and -7 to get 14.

$$\frac{-2x^{2}+\frac{1}{4}x\sqrt{97}+\frac{14}{8}x-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Reduce the fraction \frac{14}{8} to lowest terms by extracting and canceling out 2.

$$\frac{-2x^{2}+\frac{1}{4}x\sqrt{97}+\frac{7}{4}x-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Combine \frac{1}{4}x\sqrt{97} and -\frac{1}{4}\sqrt{97}x to get 0.

$$\frac{-2x^{2}+\frac{7}{4}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Express -\frac{1}{4}\times 97 as a single fraction.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{-97}{4}\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Fraction \frac{-97}{4} can be rewritten as -\frac{97}{4} by extracting the negative sign.

$$\frac{-2x^{2}+\frac{7}{4}x-\frac{97}{4}\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply -\frac{97}{4} times -\frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{-97\left(-1\right)}{4\times 8}-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Do the multiplications in the fraction \frac{-97\left(-1\right)}{4\times 8}.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{97}{32}-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply -\frac{1}{4} times -\frac{7}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{97}{32}+\frac{-\left(-7\right)}{4\times 8}\sqrt{97}+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Do the multiplications in the fraction \frac{-\left(-7\right)}{4\times 8}.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Combine \frac{7}{4}x and \frac{7}{4}x to get \frac{7}{2}x.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply \frac{7}{4} times -\frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}+\frac{7\left(-1\right)}{4\times 8}\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Do the multiplications in the fraction \frac{7\left(-1\right)}{4\times 8}.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}+\frac{-7}{32}\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Fraction \frac{-7}{32} can be rewritten as -\frac{7}{32} by extracting the negative sign.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}-\frac{7}{32}\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Combine \frac{7}{32}\sqrt{97} and -\frac{7}{32}\sqrt{97} to get 0.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply \frac{7}{4} times -\frac{7}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7\left(-7\right)}{4\times 8}}{x}$$

Do the multiplications in the fraction \frac{7\left(-7\right)}{4\times 8}.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{-49}{32}}{x}$$

Fraction \frac{-49}{32} can be rewritten as -\frac{49}{32} by extracting the negative sign.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}-\frac{49}{32}}{x}$$

Since \frac{97}{32} and \frac{49}{32} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97-49}{32}}{x}$$

Subtract 49 from 97 to get 48.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{48}{32}}{x}$$

Reduce the fraction \frac{48}{32} to lowest terms by extracting and canceling out 16.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{3}{2}}{x}$$

Expand

$$-2x+\frac{7}{2}+\frac{3}{2x}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{6x+6}{4x}.

$$\frac{6\left(x+1\right)}{4x}-6+8+x\left(-2\right)$$

Cancel out 2 in both numerator and denominator.

$$\frac{3\left(x+1\right)}{2x}-6+8+x\left(-2\right)$$

Add -6 and 8 to get 2.

$$\frac{3\left(x+1\right)}{2x}+2+x\left(-2\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2+x\left(-2\right) times \frac{2x}{2x}.

$$\frac{3\left(x+1\right)}{2x}+\frac{\left(2+x\left(-2\right)\right)\times 2x}{2x}$$

Since \frac{3\left(x+1\right)}{2x} and \frac{\left(2+x\left(-2\right)\right)\times 2x}{2x} have the same denominator, add them by adding their numerators.

$$\frac{3\left(x+1\right)+\left(2+x\left(-2\right)\right)\times 2x}{2x}$$

Do the multiplications in 3\left(x+1\right)+\left(2+x\left(-2\right)\right)\times 2x.

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

Combine like terms in 3x+3+4x-4x^{2}.

$$\frac{7x+3-4x^{2}}{2x}$$

Factor the expressions that are not already factored in \frac{7x+3-4x^{2}}{2x}.

$$\frac{-4\left(x-\left(-\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)}{2x}$$

Cancel out 2 in both numerator and denominator.

$$\frac{-2\left(x-\left(-\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)}{x}$$

To find the opposite of -\frac{1}{8}\sqrt{97}+\frac{7}{8}, find the opposite of each term.

$$\frac{-2\left(x-\left(-\frac{1}{8}\sqrt{97}\right)-\frac{7}{8}\right)\left(x-\left(\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)}{x}$$

The opposite of -\frac{1}{8}\sqrt{97} is \frac{1}{8}\sqrt{97}.

$$\frac{-2\left(x+\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)\left(x-\left(\frac{1}{8}\sqrt{97}+\frac{7}{8}\right)\right)}{x}$$

To find the opposite of \frac{1}{8}\sqrt{97}+\frac{7}{8}, find the opposite of each term.

$$\frac{-2\left(x+\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Use the distributive property to multiply -2 by x+\frac{1}{8}\sqrt{97}-\frac{7}{8}.

$$\frac{\left(-2x-2\times \frac{1}{8}\sqrt{97}-2\left(-\frac{7}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Multiply -2 and \frac{1}{8} to get \frac{-2}{8}.

$$\frac{\left(-2x+\frac{-2}{8}\sqrt{97}-2\left(-\frac{7}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Reduce the fraction \frac{-2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{\left(-2x-\frac{1}{4}\sqrt{97}-2\left(-\frac{7}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Express -2\left(-\frac{7}{8}\right) as a single fraction.

$$\frac{\left(-2x-\frac{1}{4}\sqrt{97}+\frac{-2\left(-7\right)}{8}\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Multiply -2 and -7 to get 14.

$$\frac{\left(-2x-\frac{1}{4}\sqrt{97}+\frac{14}{8}\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Reduce the fraction \frac{14}{8} to lowest terms by extracting and canceling out 2.

$$\frac{\left(-2x-\frac{1}{4}\sqrt{97}+\frac{7}{4}\right)\left(x-\frac{1}{8}\sqrt{97}-\frac{7}{8}\right)}{x}$$

Apply the distributive property by multiplying each term of -2x-\frac{1}{4}\sqrt{97}+\frac{7}{4} by each term of x-\frac{1}{8}\sqrt{97}-\frac{7}{8}.

$$\frac{-2x^{2}-2x\left(-\frac{1}{8}\right)\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\sqrt{97}\left(-\frac{1}{8}\right)\sqrt{97}-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply \sqrt{97} and \sqrt{97} to get 97.

$$\frac{-2x^{2}-2x\left(-\frac{1}{8}\right)\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Express -2\left(-\frac{1}{8}\right) as a single fraction.

$$\frac{-2x^{2}+\frac{-2\left(-1\right)}{8}x\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply -2 and -1 to get 2.

$$\frac{-2x^{2}+\frac{2}{8}x\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{-2x^{2}+\frac{1}{4}x\sqrt{97}-2x\left(-\frac{7}{8}\right)-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Express -2\left(-\frac{7}{8}\right) as a single fraction.

$$\frac{-2x^{2}+\frac{1}{4}x\sqrt{97}+\frac{-2\left(-7\right)}{8}x-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply -2 and -7 to get 14.

$$\frac{-2x^{2}+\frac{1}{4}x\sqrt{97}+\frac{14}{8}x-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Reduce the fraction \frac{14}{8} to lowest terms by extracting and canceling out 2.

$$\frac{-2x^{2}+\frac{1}{4}x\sqrt{97}+\frac{7}{4}x-\frac{1}{4}\sqrt{97}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Combine \frac{1}{4}x\sqrt{97} and -\frac{1}{4}\sqrt{97}x to get 0.

$$\frac{-2x^{2}+\frac{7}{4}x-\frac{1}{4}\times 97\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Express -\frac{1}{4}\times 97 as a single fraction.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{-97}{4}\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Fraction \frac{-97}{4} can be rewritten as -\frac{97}{4} by extracting the negative sign.

$$\frac{-2x^{2}+\frac{7}{4}x-\frac{97}{4}\left(-\frac{1}{8}\right)-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply -\frac{97}{4} times -\frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{-97\left(-1\right)}{4\times 8}-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Do the multiplications in the fraction \frac{-97\left(-1\right)}{4\times 8}.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{97}{32}-\frac{1}{4}\sqrt{97}\left(-\frac{7}{8}\right)+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply -\frac{1}{4} times -\frac{7}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{97}{32}+\frac{-\left(-7\right)}{4\times 8}\sqrt{97}+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Do the multiplications in the fraction \frac{-\left(-7\right)}{4\times 8}.

$$\frac{-2x^{2}+\frac{7}{4}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}+\frac{7}{4}x+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Combine \frac{7}{4}x and \frac{7}{4}x to get \frac{7}{2}x.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}+\frac{7}{4}\left(-\frac{1}{8}\right)\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply \frac{7}{4} times -\frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}+\frac{7\left(-1\right)}{4\times 8}\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Do the multiplications in the fraction \frac{7\left(-1\right)}{4\times 8}.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}+\frac{-7}{32}\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Fraction \frac{-7}{32} can be rewritten as -\frac{7}{32} by extracting the negative sign.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{32}\sqrt{97}-\frac{7}{32}\sqrt{97}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Combine \frac{7}{32}\sqrt{97} and -\frac{7}{32}\sqrt{97} to get 0.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7}{4}\left(-\frac{7}{8}\right)}{x}$$

Multiply \frac{7}{4} times -\frac{7}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{7\left(-7\right)}{4\times 8}}{x}$$

Do the multiplications in the fraction \frac{7\left(-7\right)}{4\times 8}.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}+\frac{-49}{32}}{x}$$

Fraction \frac{-49}{32} can be rewritten as -\frac{49}{32} by extracting the negative sign.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97}{32}-\frac{49}{32}}{x}$$

Since \frac{97}{32} and \frac{49}{32} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{97-49}{32}}{x}$$

Subtract 49 from 97 to get 48.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{48}{32}}{x}$$

Reduce the fraction \frac{48}{32} to lowest terms by extracting and canceling out 16.

$$\frac{-2x^{2}+\frac{7}{2}x+\frac{3}{2}}{x}$$