Let's solve the equation:

\[
\frac{1}{r-2} - \frac{1}{r^2 - 7r + 10} = \frac{6}{r-2}
\]

### Step 1: Factor the quadratic expression

First, let's factor the quadratic expression \( r^2 - 7r + 10 \).

To factor this, we need two numbers that multiply to 10 and add to -7. The factors are -5 and -2:

\[
r^2 - 7r + 10 = (r-5)(r-2)
\]

### Step 2: Substitute the factored form

Substitute the factored form into the equation:

\[
\frac{1}{r-2} - \frac{1}{(r-5)(r-2)} = \frac{6}{r-2}
\]

### Step 3: Eliminate the denominators

Multiply every term by the common denominator \((r-5)(r-2)\) to eliminate the fractions:

\[
\frac{(r-5)(r-2)}{r-2} - \frac{(r-5)(r-2)}{(r-5)(r-2)} = \frac{6(r-5)(r-2)}{r-2}
\]

Simplify each term:

\[
r-5 - 1 = 6(r-5)
\]

### Step 4: Solve the resulting linear equation

Now simplify and solve the equation:

\[
r - 5 - 1 = 6r - 30
\]

Combine like terms:

\[
r - 6 = 6r - 30
\]

Subtract \( r \) from both sides:

\[
-6 = 5r - 30
\]

Add 30 to both sides:

\[
24 = 5r
\]

Divide by 5:

\[
r = \frac{24}{5} = 4.8
\]

### Step 5: Check the solution

Substitute \( r = 4.8 \) back into the original equation to ensure it satisfies the equation.

However, note that \( r \) cannot be equal to 2 or 5 because these values would make the denominators zero, which is undefined. Since \( r = 4.8 \) does not make any denominator zero, it is a valid solution.

**Thus, the solution is \( r = 4.8 \).**

Would you like to explore any other steps in detail or see a different approach?

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Here are 5 related questions for further exploration:

1. How do you solve equations involving rational expressions?
2. Why is it important to check for undefined values in rational equations?
3. Can you factor a quadratic expression in general form?
4. What are the steps to solve an equation by clearing fractions?
5. How do you solve quadratic equations that are set equal to zero?

**Tip:** When working with rational equations, always check the solutions for any restrictions, such as values that make the denominator zero.

Learn how to solve the rational equation 1/(r-2) - 1/(r²-7x+10) = 6/(r-2) with a step-by-step solution. This problem involves rational expressions and quadratic equations, exploring methods to solve complex algebraic equations. Suitable for high school students.

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