Solve (2x - 6)/(x ^ 2 - 9x + 20) - (x - 1)/(x ^ 2 - 7x + 12) - (x - 2)/(x ^ 2 - 8x + 15) | 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

$$\frac{2x-6}{x^{2}-9x+20}-\frac{x-1}{x^{2}-7x+12}-\frac{x-2}{x^{2}-8x+15}$$

Evaluate

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

Short Solution Steps

Factor $x^{2}-9x+20$. Factor $x^{2}-7x+12$.

$$\frac{2x-6}{\left(x-5\right)\left(x-4\right)}-\frac{x-1}{\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-5\right)\left(x-4\right)$ and $\left(x-4\right)\left(x-3\right)$ is $\left(x-5\right)\left(x-4\right)\left(x-3\right)$. Multiply $\frac{2x-6}{\left(x-5\right)\left(x-4\right)}$ times $\frac{x-3}{x-3}$. Multiply $\frac{x-1}{\left(x-4\right)\left(x-3\right)}$ times $\frac{x-5}{x-5}$.

$$\frac{\left(2x-6\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{\left(x-1\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

Since $\frac{\left(2x-6\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ and $\frac{\left(x-1\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x-6\right)\left(x-3\right)-\left(x-1\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

Do the multiplications in $\left(2x-6\right)\left(x-3\right)-\left(x-1\right)\left(x-5\right)$.

$$\frac{2x^{2}-6x-6x+18-x^{2}+5x+x-5}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

Combine like terms in $2x^{2}-6x-6x+18-x^{2}+5x+x-5$.

$$\frac{x^{2}-6x+13}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

Factor $x^{2}-8x+15$.

$$\frac{x^{2}-6x+13}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{\left(x-5\right)\left(x-3\right)}$$

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

$$\frac{x^{2}-6x+13}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{\left(x-2\right)\left(x-4\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Since $\frac{x^{2}-6x+13}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ and $\frac{\left(x-2\right)\left(x-4\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-6x+13-\left(x-2\right)\left(x-4\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Do the multiplications in $x^{2}-6x+13-\left(x-2\right)\left(x-4\right)$.

$$\frac{x^{2}-6x+13-x^{2}+4x+2x-8}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Combine like terms in $x^{2}-6x+13-x^{2}+4x+2x-8$.

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

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

$$\frac{5}{x^{3}-12x^{2}+47x-60}$$

Expand

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

Short Solution Steps

Factor $x^{2}-9x+20$. Factor $x^{2}-7x+12$.

$$\frac{2x-6}{\left(x-5\right)\left(x-4\right)}-\frac{x-1}{\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-5\right)\left(x-4\right)$ and $\left(x-4\right)\left(x-3\right)$ is $\left(x-5\right)\left(x-4\right)\left(x-3\right)$. Multiply $\frac{2x-6}{\left(x-5\right)\left(x-4\right)}$ times $\frac{x-3}{x-3}$. Multiply $\frac{x-1}{\left(x-4\right)\left(x-3\right)}$ times $\frac{x-5}{x-5}$.

$$\frac{\left(2x-6\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{\left(x-1\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

Since $\frac{\left(2x-6\right)\left(x-3\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ and $\frac{\left(x-1\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x-6\right)\left(x-3\right)-\left(x-1\right)\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

Do the multiplications in $\left(2x-6\right)\left(x-3\right)-\left(x-1\right)\left(x-5\right)$.

$$\frac{2x^{2}-6x-6x+18-x^{2}+5x+x-5}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

Combine like terms in $2x^{2}-6x-6x+18-x^{2}+5x+x-5$.

$$\frac{x^{2}-6x+13}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x^{2}-8x+15}$$

Factor $x^{2}-8x+15$.

$$\frac{x^{2}-6x+13}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{x-2}{\left(x-5\right)\left(x-3\right)}$$

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

$$\frac{x^{2}-6x+13}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}-\frac{\left(x-2\right)\left(x-4\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Since $\frac{x^{2}-6x+13}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ and $\frac{\left(x-2\right)\left(x-4\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-6x+13-\left(x-2\right)\left(x-4\right)}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Do the multiplications in $x^{2}-6x+13-\left(x-2\right)\left(x-4\right)$.

$$\frac{x^{2}-6x+13-x^{2}+4x+2x-8}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$$

Combine like terms in $x^{2}-6x+13-x^{2}+4x+2x-8$.

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

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

$$\frac{5}{x^{3}-12x^{2}+47x-60}$$