Solve (22 - (- 14))/(6x - 7) - (4 ^ 2 * x - 6 - 18)/(72/8 * 3) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{22-(-14)}{6x-7}-\frac{4^{2}x-6-18}{72\div8\times3}$$

Evaluate

$\frac{4\left(201+64x-24x^{2}\right)}{27\left(6x-7\right)}$

Short Solution Steps

The opposite of $-14$ is $14$.

$$\frac{22+14}{6x-7}-\frac{4^{2}x-6-18}{\frac{72}{8}\times 3}$$

Add $22$ and $14$ to get $36$.

$$\frac{36}{6x-7}-\frac{4^{2}x-6-18}{\frac{72}{8}\times 3}$$

Calculate $4$ to the power of $2$ and get $16$.

$$\frac{36}{6x-7}-\frac{16x-6-18}{\frac{72}{8}\times 3}$$

Subtract $18$ from $-6$ to get $-24$.

$$\frac{36}{6x-7}-\frac{16x-24}{\frac{72}{8}\times 3}$$

Divide $72$ by $8$ to get $9$.

$$\frac{36}{6x-7}-\frac{16x-24}{9\times 3}$$

Multiply $9$ and $3$ to get $27$.

$$\frac{36}{6x-7}-\frac{16x-24}{27}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $6x-7$ and $27$ is $27\left(6x-7\right)$. Multiply $\frac{36}{6x-7}$ times $\frac{27}{27}$. Multiply $\frac{16x-24}{27}$ times $\frac{6x-7}{6x-7}$.

$$\frac{36\times 27}{27\left(6x-7\right)}-\frac{\left(16x-24\right)\left(6x-7\right)}{27\left(6x-7\right)}$$

Since $\frac{36\times 27}{27\left(6x-7\right)}$ and $\frac{\left(16x-24\right)\left(6x-7\right)}{27\left(6x-7\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{36\times 27-\left(16x-24\right)\left(6x-7\right)}{27\left(6x-7\right)}$$

Do the multiplications in $36\times 27-\left(16x-24\right)\left(6x-7\right)$.

$$\frac{972-96x^{2}+112x+144x-168}{27\left(6x-7\right)}$$

Combine like terms in $972-96x^{2}+112x+144x-168$.

$$\frac{804-96x^{2}+256x}{27\left(6x-7\right)}$$

Factor the expressions that are not already factored in $\frac{804-96x^{2}+256x}{27\left(6x-7\right)}$.

$$\frac{-4\times 24\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{27\left(6x-7\right)}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{-4\times 8\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{9\left(6x-7\right)}$$

Expand $9\left(6x-7\right)$.

$$\frac{-4\times 8\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{54x-63}$$

Multiply $-4$ and $8$ to get $-32$.

$$\frac{-32\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{54x-63}$$

To find the opposite of $-\frac{1}{12}\sqrt{1462}+\frac{4}{3}$, find the opposite of each term.

$$\frac{-32\left(x-\left(-\frac{1}{12}\sqrt{1462}\right)-\frac{4}{3}\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{54x-63}$$

The opposite of $-\frac{1}{12}\sqrt{1462}$ is $\frac{1}{12}\sqrt{1462}$.

$$\frac{-32\left(x+\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{54x-63}$$

To find the opposite of $\frac{1}{12}\sqrt{1462}+\frac{4}{3}$, find the opposite of each term.

$$\frac{-32\left(x+\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Use the distributive property to multiply $-32$ by $x+\frac{1}{12}\sqrt{1462}-\frac{4}{3}$.

$$\frac{\left(-32x-32\times \frac{1}{12}\sqrt{1462}-32\left(-\frac{4}{3}\right)\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Multiply $-32$ and $\frac{1}{12}$ to get $\frac{-32}{12}$.

$$\frac{\left(-32x+\frac{-32}{12}\sqrt{1462}-32\left(-\frac{4}{3}\right)\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Reduce the fraction $\frac{-32}{12}$ to lowest terms by extracting and canceling out $4$.

$$\frac{\left(-32x-\frac{8}{3}\sqrt{1462}-32\left(-\frac{4}{3}\right)\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Express $-32\left(-\frac{4}{3}\right)$ as a single fraction.

$$\frac{\left(-32x-\frac{8}{3}\sqrt{1462}+\frac{-32\left(-4\right)}{3}\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Multiply $-32$ and $-4$ to get $128$.

$$\frac{\left(-32x-\frac{8}{3}\sqrt{1462}+\frac{128}{3}\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Apply the distributive property by multiplying each term of $-32x-\frac{8}{3}\sqrt{1462}+\frac{128}{3}$ by each term of $x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}$.

$$\frac{-32x^{2}-32x\left(-\frac{1}{12}\right)\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\sqrt{1462}\left(-\frac{1}{12}\right)\sqrt{1462}-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $\sqrt{1462}$ and $\sqrt{1462}$ to get $1462$.

$$\frac{-32x^{2}-32x\left(-\frac{1}{12}\right)\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Express $-32\left(-\frac{1}{12}\right)$ as a single fraction.

$$\frac{-32x^{2}+\frac{-32\left(-1\right)}{12}x\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-32$ and $-1$ to get $32$.

$$\frac{-32x^{2}+\frac{32}{12}x\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Reduce the fraction $\frac{32}{12}$ to lowest terms by extracting and canceling out $4$.

$$\frac{-32x^{2}+\frac{8}{3}x\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Express $-32\left(-\frac{4}{3}\right)$ as a single fraction.

$$\frac{-32x^{2}+\frac{8}{3}x\sqrt{1462}+\frac{-32\left(-4\right)}{3}x-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-32$ and $-4$ to get $128$.

$$\frac{-32x^{2}+\frac{8}{3}x\sqrt{1462}+\frac{128}{3}x-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Combine $\frac{8}{3}x\sqrt{1462}$ and $-\frac{8}{3}\sqrt{1462}x$ to get $0$.

$$\frac{-32x^{2}+\frac{128}{3}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Express $-\frac{8}{3}\times 1462$ as a single fraction.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{-8\times 1462}{3}\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-8$ and $1462$ to get $-11696$.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{-11696}{3}\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Fraction $\frac{-11696}{3}$ can be rewritten as $-\frac{11696}{3}$ by extracting the negative sign.

$$\frac{-32x^{2}+\frac{128}{3}x-\frac{11696}{3}\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-\frac{11696}{3}$ times $-\frac{1}{12}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{-11696\left(-1\right)}{3\times 12}-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Do the multiplications in the fraction $\frac{-11696\left(-1\right)}{3\times 12}$.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{11696}{36}-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Reduce the fraction $\frac{11696}{36}$ to lowest terms by extracting and canceling out $4$.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{2924}{9}-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-\frac{8}{3}$ times $-\frac{4}{3}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{2924}{9}+\frac{-8\left(-4\right)}{3\times 3}\sqrt{1462}+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Do the multiplications in the fraction $\frac{-8\left(-4\right)}{3\times 3}$.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Combine $\frac{128}{3}x$ and $\frac{128}{3}x$ to get $\frac{256}{3}x$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $\frac{128}{3}$ times $-\frac{1}{12}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}+\frac{128\left(-1\right)}{3\times 12}\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Do the multiplications in the fraction $\frac{128\left(-1\right)}{3\times 12}$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}+\frac{-128}{36}\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Reduce the fraction $\frac{-128}{36}$ to lowest terms by extracting and canceling out $4$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}-\frac{32}{9}\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Combine $\frac{32}{9}\sqrt{1462}$ and $-\frac{32}{9}\sqrt{1462}$ to get $0$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $\frac{128}{3}$ times $-\frac{4}{3}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{128\left(-4\right)}{3\times 3}}{54x-63}$$

Do the multiplications in the fraction $\frac{128\left(-4\right)}{3\times 3}$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{-512}{9}}{54x-63}$$

Fraction $\frac{-512}{9}$ can be rewritten as $-\frac{512}{9}$ by extracting the negative sign.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}-\frac{512}{9}}{54x-63}$$

Since $\frac{2924}{9}$ and $\frac{512}{9}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924-512}{9}}{54x-63}$$

Subtract $512$ from $2924$ to get $2412$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2412}{9}}{54x-63}$$

Divide $2412$ by $9$ to get $268$.

$$\frac{-32x^{2}+\frac{256}{3}x+268}{54x-63}$$

Factor the expressions that are not already factored.

$$\frac{-\frac{4}{3}\times 24\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{9\left(6x-7\right)}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{-\frac{4}{3}\times 8\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{3\left(6x-7\right)}$$

Expand the expression.

$$\frac{-\frac{32}{3}x^{2}+\frac{256}{9}x+\frac{268}{3}}{18x-21}$$

Factor the expressions that are not already factored.

$$\frac{-\frac{4}{9}\times 24\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{3\left(6x-7\right)}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{-\frac{4}{9}\times 8\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{6x-7}$$

Expand the expression.

$$\frac{-\frac{32}{9}x^{2}+\frac{256}{27}x+\frac{268}{9}}{6x-7}$$

Expand

$-\frac{4\left(24x^{2}-64x-201\right)}{27\left(6x-7\right)}$

Short Solution Steps

The opposite of $-14$ is $14$.

$$\frac{22+14}{6x-7}-\frac{4^{2}x-6-18}{\frac{72}{8}\times 3}$$

Add $22$ and $14$ to get $36$.

$$\frac{36}{6x-7}-\frac{4^{2}x-6-18}{\frac{72}{8}\times 3}$$

Calculate $4$ to the power of $2$ and get $16$.

$$\frac{36}{6x-7}-\frac{16x-6-18}{\frac{72}{8}\times 3}$$

Subtract $18$ from $-6$ to get $-24$.

$$\frac{36}{6x-7}-\frac{16x-24}{\frac{72}{8}\times 3}$$

Divide $72$ by $8$ to get $9$.

$$\frac{36}{6x-7}-\frac{16x-24}{9\times 3}$$

Multiply $9$ and $3$ to get $27$.

$$\frac{36}{6x-7}-\frac{16x-24}{27}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $6x-7$ and $27$ is $27\left(6x-7\right)$. Multiply $\frac{36}{6x-7}$ times $\frac{27}{27}$. Multiply $\frac{16x-24}{27}$ times $\frac{6x-7}{6x-7}$.

$$\frac{36\times 27}{27\left(6x-7\right)}-\frac{\left(16x-24\right)\left(6x-7\right)}{27\left(6x-7\right)}$$

Since $\frac{36\times 27}{27\left(6x-7\right)}$ and $\frac{\left(16x-24\right)\left(6x-7\right)}{27\left(6x-7\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{36\times 27-\left(16x-24\right)\left(6x-7\right)}{27\left(6x-7\right)}$$

Do the multiplications in $36\times 27-\left(16x-24\right)\left(6x-7\right)$.

$$\frac{972-96x^{2}+112x+144x-168}{27\left(6x-7\right)}$$

Combine like terms in $972-96x^{2}+112x+144x-168$.

$$\frac{804-96x^{2}+256x}{27\left(6x-7\right)}$$

Factor the expressions that are not already factored in $\frac{804-96x^{2}+256x}{27\left(6x-7\right)}$.

$$\frac{-4\times 24\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{27\left(6x-7\right)}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{-4\times 8\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{9\left(6x-7\right)}$$

Expand $9\left(6x-7\right)$.

$$\frac{-4\times 8\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{54x-63}$$

Multiply $-4$ and $8$ to get $-32$.

$$\frac{-32\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{54x-63}$$

To find the opposite of $-\frac{1}{12}\sqrt{1462}+\frac{4}{3}$, find the opposite of each term.

$$\frac{-32\left(x-\left(-\frac{1}{12}\sqrt{1462}\right)-\frac{4}{3}\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{54x-63}$$

The opposite of $-\frac{1}{12}\sqrt{1462}$ is $\frac{1}{12}\sqrt{1462}$.

$$\frac{-32\left(x+\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{54x-63}$$

To find the opposite of $\frac{1}{12}\sqrt{1462}+\frac{4}{3}$, find the opposite of each term.

$$\frac{-32\left(x+\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Use the distributive property to multiply $-32$ by $x+\frac{1}{12}\sqrt{1462}-\frac{4}{3}$.

$$\frac{\left(-32x-32\times \frac{1}{12}\sqrt{1462}-32\left(-\frac{4}{3}\right)\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Multiply $-32$ and $\frac{1}{12}$ to get $\frac{-32}{12}$.

$$\frac{\left(-32x+\frac{-32}{12}\sqrt{1462}-32\left(-\frac{4}{3}\right)\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Reduce the fraction $\frac{-32}{12}$ to lowest terms by extracting and canceling out $4$.

$$\frac{\left(-32x-\frac{8}{3}\sqrt{1462}-32\left(-\frac{4}{3}\right)\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Express $-32\left(-\frac{4}{3}\right)$ as a single fraction.

$$\frac{\left(-32x-\frac{8}{3}\sqrt{1462}+\frac{-32\left(-4\right)}{3}\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Multiply $-32$ and $-4$ to get $128$.

$$\frac{\left(-32x-\frac{8}{3}\sqrt{1462}+\frac{128}{3}\right)\left(x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}\right)}{54x-63}$$

Apply the distributive property by multiplying each term of $-32x-\frac{8}{3}\sqrt{1462}+\frac{128}{3}$ by each term of $x-\frac{1}{12}\sqrt{1462}-\frac{4}{3}$.

$$\frac{-32x^{2}-32x\left(-\frac{1}{12}\right)\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\sqrt{1462}\left(-\frac{1}{12}\right)\sqrt{1462}-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $\sqrt{1462}$ and $\sqrt{1462}$ to get $1462$.

$$\frac{-32x^{2}-32x\left(-\frac{1}{12}\right)\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Express $-32\left(-\frac{1}{12}\right)$ as a single fraction.

$$\frac{-32x^{2}+\frac{-32\left(-1\right)}{12}x\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-32$ and $-1$ to get $32$.

$$\frac{-32x^{2}+\frac{32}{12}x\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Reduce the fraction $\frac{32}{12}$ to lowest terms by extracting and canceling out $4$.

$$\frac{-32x^{2}+\frac{8}{3}x\sqrt{1462}-32x\left(-\frac{4}{3}\right)-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Express $-32\left(-\frac{4}{3}\right)$ as a single fraction.

$$\frac{-32x^{2}+\frac{8}{3}x\sqrt{1462}+\frac{-32\left(-4\right)}{3}x-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-32$ and $-4$ to get $128$.

$$\frac{-32x^{2}+\frac{8}{3}x\sqrt{1462}+\frac{128}{3}x-\frac{8}{3}\sqrt{1462}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Combine $\frac{8}{3}x\sqrt{1462}$ and $-\frac{8}{3}\sqrt{1462}x$ to get $0$.

$$\frac{-32x^{2}+\frac{128}{3}x-\frac{8}{3}\times 1462\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Express $-\frac{8}{3}\times 1462$ as a single fraction.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{-8\times 1462}{3}\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-8$ and $1462$ to get $-11696$.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{-11696}{3}\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Fraction $\frac{-11696}{3}$ can be rewritten as $-\frac{11696}{3}$ by extracting the negative sign.

$$\frac{-32x^{2}+\frac{128}{3}x-\frac{11696}{3}\left(-\frac{1}{12}\right)-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-\frac{11696}{3}$ times $-\frac{1}{12}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{-11696\left(-1\right)}{3\times 12}-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Do the multiplications in the fraction $\frac{-11696\left(-1\right)}{3\times 12}$.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{11696}{36}-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Reduce the fraction $\frac{11696}{36}$ to lowest terms by extracting and canceling out $4$.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{2924}{9}-\frac{8}{3}\sqrt{1462}\left(-\frac{4}{3}\right)+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $-\frac{8}{3}$ times $-\frac{4}{3}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{2924}{9}+\frac{-8\left(-4\right)}{3\times 3}\sqrt{1462}+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Do the multiplications in the fraction $\frac{-8\left(-4\right)}{3\times 3}$.

$$\frac{-32x^{2}+\frac{128}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}+\frac{128}{3}x+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Combine $\frac{128}{3}x$ and $\frac{128}{3}x$ to get $\frac{256}{3}x$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}+\frac{128}{3}\left(-\frac{1}{12}\right)\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $\frac{128}{3}$ times $-\frac{1}{12}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}+\frac{128\left(-1\right)}{3\times 12}\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Do the multiplications in the fraction $\frac{128\left(-1\right)}{3\times 12}$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}+\frac{-128}{36}\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Reduce the fraction $\frac{-128}{36}$ to lowest terms by extracting and canceling out $4$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{32}{9}\sqrt{1462}-\frac{32}{9}\sqrt{1462}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Combine $\frac{32}{9}\sqrt{1462}$ and $-\frac{32}{9}\sqrt{1462}$ to get $0$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{128}{3}\left(-\frac{4}{3}\right)}{54x-63}$$

Multiply $\frac{128}{3}$ times $-\frac{4}{3}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{128\left(-4\right)}{3\times 3}}{54x-63}$$

Do the multiplications in the fraction $\frac{128\left(-4\right)}{3\times 3}$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}+\frac{-512}{9}}{54x-63}$$

Fraction $\frac{-512}{9}$ can be rewritten as $-\frac{512}{9}$ by extracting the negative sign.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924}{9}-\frac{512}{9}}{54x-63}$$

Since $\frac{2924}{9}$ and $\frac{512}{9}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2924-512}{9}}{54x-63}$$

Subtract $512$ from $2924$ to get $2412$.

$$\frac{-32x^{2}+\frac{256}{3}x+\frac{2412}{9}}{54x-63}$$

Divide $2412$ by $9$ to get $268$.

$$\frac{-32x^{2}+\frac{256}{3}x+268}{54x-63}$$

Factor the expressions that are not already factored.

$$\frac{-\frac{4}{3}\times 24\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{9\left(6x-7\right)}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{-\frac{4}{3}\times 8\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{3\left(6x-7\right)}$$

Expand the expression.

$$\frac{-\frac{32}{3}x^{2}+\frac{256}{9}x+\frac{268}{3}}{18x-21}$$

Factor the expressions that are not already factored.

$$\frac{-\frac{4}{9}\times 24\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{3\left(6x-7\right)}$$

Cancel out $3$ in both numerator and denominator.

$$\frac{-\frac{4}{9}\times 8\left(x-\left(-\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{1462}+\frac{4}{3}\right)\right)}{6x-7}$$

Expand the expression.

$$\frac{-\frac{32}{9}x^{2}+\frac{256}{27}x+\frac{268}{9}}{6x-7}$$