Solve 1/(x-3)(x-4)+1/(x-4)(x-5)+1/(x-5)(x-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

$$1/(x-3)(x-4)+1/(x-4)(x-5)+1/(x-5)(x-3)$$

Evaluate

$\frac{3x^{3}-36x^{2}+144x-191}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$

Short Solution Steps

Express $\frac{1}{x-3}\left(x-4\right)$ as a single fraction.

$$\frac{x-4}{x-3}+\frac{1}{x-4}\left(x-5\right)+\frac{1}{x-5}\left(x-3\right)$$

Express $\frac{1}{x-4}\left(x-5\right)$ as a single fraction.

$$\frac{x-4}{x-3}+\frac{x-5}{x-4}+\frac{1}{x-5}\left(x-3\right)$$

Express $\frac{1}{x-5}\left(x-3\right)$ as a single fraction.

$$\frac{x-4}{x-3}+\frac{x-5}{x-4}+\frac{x-3}{x-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-3$ and $x-4$ is $\left(x-4\right)\left(x-3\right)$. Multiply $\frac{x-4}{x-3}$ times $\frac{x-4}{x-4}$. Multiply $\frac{x-5}{x-4}$ times $\frac{x-3}{x-3}$.

$$\frac{\left(x-4\right)\left(x-4\right)}{\left(x-4\right)\left(x-3\right)}+\frac{\left(x-5\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}+\frac{x-3}{x-5}$$

Since $\frac{\left(x-4\right)\left(x-4\right)}{\left(x-4\right)\left(x-3\right)}$ and $\frac{\left(x-5\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(x-4\right)\left(x-4\right)+\left(x-5\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}+\frac{x-3}{x-5}$$

Do the multiplications in $\left(x-4\right)\left(x-4\right)+\left(x-5\right)\left(x-3\right)$.

$$\frac{x^{2}-4x-4x+16+x^{2}-3x-5x+15}{\left(x-4\right)\left(x-3\right)}+\frac{x-3}{x-5}$$

Combine like terms in $x^{2}-4x-4x+16+x^{2}-3x-5x+15$.

$$\frac{2x^{2}-16x+31}{\left(x-4\right)\left(x-3\right)}+\frac{x-3}{x-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-4\right)\left(x-3\right)$ and $x-5$ is $\left(x-5\right)\left(x-4\right)\left(x-3\right)$. Multiply $\frac{2x^{2}-16x+31}{\left(x-4\right)\left(x-3\right)}$ times $\frac{x-5}{x-5}$. Multiply $\frac{x-3}{x-5}$ times $\frac{\left(x-4\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}$.

$$\frac{\left(2x^{2}-16x+31\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}+\frac{\left(x-3\right)\left(x-4\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Since $\frac{\left(2x^{2}-16x+31\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ and $\frac{\left(x-3\right)\left(x-4\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(2x^{2}-16x+31\right)\left(x-5\right)+\left(x-3\right)\left(x-4\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Do the multiplications in $\left(2x^{2}-16x+31\right)\left(x-5\right)+\left(x-3\right)\left(x-4\right)\left(x-3\right)$.

$$\frac{2x^{3}-10x^{2}-16x^{2}+80x+31x-155+x^{3}-7x^{2}+12x-3x^{2}+21x-36}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Combine like terms in $2x^{3}-10x^{2}-16x^{2}+80x+31x-155+x^{3}-7x^{2}+12x-3x^{2}+21x-36$.

$$\frac{3x^{3}-36x^{2}+144x-191}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Expand $\left(x-5\right)\left(x-4\right)\left(x-3\right)$.

$$\frac{3x^{3}-36x^{2}+144x-191}{x^{3}-12x^{2}+47x-60}$$

Expand

$\frac{3x^{3}-36x^{2}+144x-191}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$

Short Solution Steps

Express $\frac{1}{x-3}\left(x-4\right)$ as a single fraction.

$$\frac{x-4}{x-3}+\frac{1}{x-4}\left(x-5\right)+\frac{1}{x-5}\left(x-3\right)$$

Express $\frac{1}{x-4}\left(x-5\right)$ as a single fraction.

$$\frac{x-4}{x-3}+\frac{x-5}{x-4}+\frac{1}{x-5}\left(x-3\right)$$

Express $\frac{1}{x-5}\left(x-3\right)$ as a single fraction.

$$\frac{x-4}{x-3}+\frac{x-5}{x-4}+\frac{x-3}{x-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-3$ and $x-4$ is $\left(x-4\right)\left(x-3\right)$. Multiply $\frac{x-4}{x-3}$ times $\frac{x-4}{x-4}$. Multiply $\frac{x-5}{x-4}$ times $\frac{x-3}{x-3}$.

$$\frac{\left(x-4\right)\left(x-4\right)}{\left(x-4\right)\left(x-3\right)}+\frac{\left(x-5\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}+\frac{x-3}{x-5}$$

Since $\frac{\left(x-4\right)\left(x-4\right)}{\left(x-4\right)\left(x-3\right)}$ and $\frac{\left(x-5\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(x-4\right)\left(x-4\right)+\left(x-5\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}+\frac{x-3}{x-5}$$

Do the multiplications in $\left(x-4\right)\left(x-4\right)+\left(x-5\right)\left(x-3\right)$.

$$\frac{x^{2}-4x-4x+16+x^{2}-3x-5x+15}{\left(x-4\right)\left(x-3\right)}+\frac{x-3}{x-5}$$

Combine like terms in $x^{2}-4x-4x+16+x^{2}-3x-5x+15$.

$$\frac{2x^{2}-16x+31}{\left(x-4\right)\left(x-3\right)}+\frac{x-3}{x-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-4\right)\left(x-3\right)$ and $x-5$ is $\left(x-5\right)\left(x-4\right)\left(x-3\right)$. Multiply $\frac{2x^{2}-16x+31}{\left(x-4\right)\left(x-3\right)}$ times $\frac{x-5}{x-5}$. Multiply $\frac{x-3}{x-5}$ times $\frac{\left(x-4\right)\left(x-3\right)}{\left(x-4\right)\left(x-3\right)}$.

$$\frac{\left(2x^{2}-16x+31\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}+\frac{\left(x-3\right)\left(x-4\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Since $\frac{\left(2x^{2}-16x+31\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ and $\frac{\left(x-3\right)\left(x-4\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(2x^{2}-16x+31\right)\left(x-5\right)+\left(x-3\right)\left(x-4\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Do the multiplications in $\left(2x^{2}-16x+31\right)\left(x-5\right)+\left(x-3\right)\left(x-4\right)\left(x-3\right)$.

$$\frac{2x^{3}-10x^{2}-16x^{2}+80x+31x-155+x^{3}-7x^{2}+12x-3x^{2}+21x-36}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Combine like terms in $2x^{3}-10x^{2}-16x^{2}+80x+31x-155+x^{3}-7x^{2}+12x-3x^{2}+21x-36$.

$$\frac{3x^{3}-36x^{2}+144x-191}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Expand $\left(x-5\right)\left(x-4\right)\left(x-3\right)$.

$$\frac{3x^{3}-36x^{2}+144x-191}{x^{3}-12x^{2}+47x-60}$$