a=1.79, b=1.83, c=0.58
(1/(a^2-ab-ac+bc))+(2/(b^2-ab-ac+bc))+(1/(c^2-ab-ac+bc))

Given the expression:

\[
\frac{1}{a^2 - ab - ac + bc} + \frac{2}{b^2 - ab - ac + bc} + \frac{1}{c^2 - ab - ac + bc}
\]

we need to evaluate it using the given values \( a = 1.79 \), \( b = 1.83 \), and \( c = 0.58 \).

Let's calculate it step by step.

### Step 1: Calculate individual terms in the denominators.
For the first term:
\[
a^2 - ab - ac + bc = (1.79)^2 - (1.79)(1.83) - (1.79)(0.58) + (1.83)(0.58)
\]

For the second term:
\[
b^2 - ab - ac + bc = (1.83)^2 - (1.79)(1.83) - (1.79)(0.58) + (1.83)(0.58)
\]

For the third term:
\[
c^2 - ab - ac + bc = (0.58)^2 - (1.79)(1.83) - (1.79)(0.58) + (1.83)(0.58)
\]

### Step 2: Plug these into the fractions and calculate each one.

Let's compute it.The value of the given expression is approximately \(-0.257\).

Would you like more details on any specific part of the calculation, or do you have any other questions?

Here are some related questions for further exploration:

1. What happens if you increase the value of \( c \)?
2. How does the expression change if you swap the values of \( a \) and \( b \)?
3. What is the significance of each term in the denominators?
4. How would the expression behave if all values were equal?
5. What is the effect of using negative values for \( a \), \( b \), or \( c \)?
6. Can this expression be simplified further analytically?
7. What happens if you set one of the variables to zero?
8. How would the result change if all values were squared first and then added or subtracted?

**Tip:** Always verify your final result by double-checking each step, especially in complex expressions.

Learn how to evaluate the fractional expression (1/(a^2 - ab - ac + bc)) + (2/(b^2 - ab - ac + bc)) + (1/(c^2 - ab - ac + bc)) with specific values a=1.79, b=1.83, c=0.58. This problem involves advanced algebraic concepts and fractional expressions. Suitable for students in Grades 11-12.

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