Solving Algebraic Expressions Using Identities

The problem asks us to evaluate a given mathematical expression:

Expression = \( \frac{(243+647)^2+(243-647)^2}{(243\times 243+647\times 647)} \)

We can simplify this expression by recognizing the pattern in the numerator and the denominator. Let's use algebraic identities to make the calculation easier.

Identifying Variables and Algebraic Identities

Let \( a = 243 \) and \( b = 647 \). The expression can be rewritten as:

Expression = \( \frac{(a+b)^2+(a-b)^2}{(a\times a+b\times b)} = \frac{(a+b)^2+(a-b)^2}{a^2+b^2} \)

Now, consider the numerator, \( (a+b)^2+(a-b)^2 \). We can expand the terms using standard algebraic identities:

• The identity for a sum squared is: \( (a+b)^2 = a^2 + 2ab + b^2 \)
• The identity for a difference squared is: \( (a-b)^2 = a^2 - 2ab + b^2 \)

Adding these two expanded forms together:

\( (a+b)^2+(a-b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) \)

\( (a+b)^2+(a-b)^2 = a^2 + 2ab + b^2 + a^2 - 2ab + b^2 \)

Notice that the \( +2ab \) and \( -2ab \) terms cancel each other out:

\( (a+b)^2+(a-b)^2 = a^2 + b^2 + a^2 + b^2 \)

\( (a+b)^2+(a-b)^2 = 2a^2 + 2b^2 \)

We can factor out a 2 from the result:

\( (a+b)^2+(a-b)^2 = 2(a^2 + b^2) \)

Simplifying the Full Expression

Now substitute this simplified form of the numerator back into the original expression:

Expression = \( \frac{2(a^2 + b^2)}{a^2+b^2} \)

Assuming that \( a^2 + b^2 \neq 0 \) (which is true since \( a=243 \) and \( b=647 \) are non-zero real numbers, so their squares are positive), we can cancel out the term \( (a^2 + b^2) \) from both the numerator and the denominator.

Expression = \( 2 \)

Conclusion on the Value of the Expression

The value of the given expression \(\frac{(243+647)^2+(243-647)^2}{(243\times 243+647\times 647)}\) simplifies to 2.

Revision Table: Key Algebraic Identities

Additional Information on Simplifying Expressions

Simplifying algebraic expressions is a fundamental skill in mathematics. It often involves using identities, factoring, and combining like terms to write an expression in its simplest form. Recognizing patterns is key.

• Recognizing Patterns: The given problem requires recognizing the structure \( \frac{(a+b)^2+(a-b)^2}{a^2+b^2} \). This structure is a strong hint to use the identity \( (a+b)^2+(a-b)^2 = 2(a^2 + b^2) \).
• Why Identities are Useful: Algebraic identities provide shortcuts. Instead of expanding \( (243+647)^2 \) and \( (243-647)^2 \) directly (which would involve large numbers), using the identity allows us to simplify the expression purely based on its structure.
• Checking Denominator: It's always good practice to consider if the denominator could be zero. In this case, \( 243^2 + 647^2 \) will definitely be a positive number (sum of two positive squares), so we don't have to worry about division by zero.

Mastering algebraic identities is crucial for solving many types of mathematical problems efficiently.