Solving and Understanding the Value of a Complex Expression: (0.9 - 0.62) / (0.92 - 2 * 0.9 * 0.6 - 0.62)

In this article, we will break down the process of solving a complex algebraic expression using clear steps and detailed explanations. The expression in question is:

(0.9 - 0.62) / (0.92 - 2 * 0.9 * 0.6 - 0.62)

Step-by-Step Solution

Let's start by breaking down the given expression into simpler components and solve it step-by-step.

1. Simplify the Numerator

The numerator of the expression is:

(0.9 - 0.62)

First, we need to calculate 0.62:

0.62 0.36

Now, subtract this result from 0.9:

0.9 - 0.36 0.54

2. Simplify the Denominator

The denominator of the expression is more complex:

(0.92 - 2 * 0.9 * 0.6 - 0.62)

First, let's calculate each term:

0.92 0.81

2 * 0.9 * 0.6 1.08

0.62 0.36

Now, combine these terms:

0.81 - 1.08 - 0.36

0.81 - 1.44 -0.63

3. Combine the Results

Now we have the simplified numerator and denominator:

0.54 / -0.63

To simplify this fraction, we can multiply the numerator and the denominator by 100 to get rid of the decimal points:

(0.54 * 100) / (-0.63 * 100) 54 / -63

Simplify the fraction:

54 / -63 -6 / 7

Therefore, the final value of the expression is:

-6 / 7 ≈ -0.857

Cross-Verification and Solution Method

In another attempt to solve the given expression, the alternative solution provided is:

(0.9 - 0.62) / (0.92 - 2 * 0.9 * 0.6 - 0.62)

This simplifies to:

(0.32) / (1.8 - 1.08 - 1.2)

This further simplifies to:

(0.6) / (0.54)

Which simplifies to:

60 / 54 60 / 54 5 / 16

This seems to be a different solution, which doesn't align with the initial steps. It appears there might be a mistake in the breakdown of the expression. Let's verify and correct it:

The correct approach is to recognize that the expression can be simplified using the formula for the difference of squares:

a2 - 2ab b2 (a - b)2

Here, a 0.9 and b 0.6. So, the denominator can be rewritten as:

(0.92 - 2 * 0.9 * 0.6 0.62

(0.9 - 0.6)2 0.32 0.09

Thus, the expression becomes:

(0.9 - 0.6) / 0.09 0.3 / 0.09 30 / 9 10 / 3

However, this step shows a different solution approach. The key here is to recognize the correct algebraic simplification based on the formula for the difference of squares.

Conclusion

In conclusion, the correct value of the expression (0.9 - 0.62) / (0.92 - 2 * 0.9 * 0.6 - 0.62) is:

-6 / 7 ≈ -0.857

This demonstrates the importance of careful algebraic manipulation and the use of fundamental formulas in solving complex expressions. Understanding these steps not only helps in solving similar problems but also in building a stronger foundation in algebra.

Keep practicing and exploring algebraic expressions to enhance your problem-solving skills. Happy computing!