Mastering the Identity: Analyzing the Equation $ x^2 - (4y^2 - 4yz + z^2) = x^2 - 4y^2 + 4yz - z^2 $

Understanding algebraic identities is fundamental in mathematics, especially in simplifying expressions and solving equations efficiently. One such intriguing identity involves rewriting and simplifying the expression:

$$
x^2 - (4y^2 - 4yz + z^2) = x^2 - 4y^2 + 4yz - z^2
$$

Understanding the Context

At first glance, the two sides appear identical but require careful analysis to reveal their deeper structure and implications. In this article, we’ll explore this equation, clarify equivalent forms, and demonstrate its applications in simplifying algebraic expressions and solving geometric or physical problems.

Step 1: Simplify the Left-Hand Side

Start by simplifying the left-hand side (LHS) of the equation:

Solved x2−4x+y2+z2=−415 (x−1)2+(y−1)2+z2=1 | Chegg.com

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Solved x2−4x+y2+z2=−415 (x−1)2+(y−1)2+z2=1 | Chegg.com
Solved x^2 - 4x + y^2 + z^2 = -15/4 x^2 + y^2 + (z + l)^2 | Chegg.com
Solved | Chegg.com
1. x2+y2+4z2=10 2. z2+4y2−4x2=4 3. 9y2+z2=16 4. | Chegg.com
x2−4y2=−zx2−z2=−yx2+16y2+4z2=1x2+4z2=yx2−y2+z2=04x2+9 | Chegg.com

Key Insights

$$
x^2 - (4y^2 - 4yz + z^2)
$$

Distributing the negative sign through the parentheses:

$$
x^2 - 4y^2 + 4yz - z^2
$$

This matches exactly with the right-hand side (RHS), confirming the algebraic identity:

$$
x^2 - (4y^2 - 4yz + z^2) = x^2 - 4y^2 + 4yz - z^2
$$

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5Question: A paleontologist discovers a sequence of five fossilized dinosaur bone lengths that form an arithmetic progression, and the sum of the first and fifth terms is 40 cm. If the third term is half the length of the second term, find the common difference of the sequence.
Solution: Let the five terms of the arithmetic sequence be $ a - 2d, a - d, a, a + d, a + 2d $. The sum of the first and fifth terms is:
The third term is $ a = 20 $. The second term is $ a - d = 20 - d $. According to the problem, the third term is half the second term:

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Final Thoughts

This equivalence demonstrates that the expression is fully simplified and symmetric in its form.

Step 2: Recognize the Structure Inside the Parentheses

The expression inside the parentheses — $4y^2 - 4yz + z^2$ — resembles a perfect square trinomial. Let’s rewrite it:

$$
4y^2 - 4yz + z^2 = (2y)^2 - 2(2y)(z) + z^2 = (2y - z)^2
$$

Thus, the original LHS becomes:

$$
x^2 - (2y - z)^2
$$

This reveals a difference of squares:
$$
x^2 - (2y - z)^2
$$

Using the identity $ a^2 - b^2 = (a - b)(a + b) $, we can rewrite:

$$
x^2 - (2y - z)^2 = (x - (2y - z))(x + (2y - z)) = (x - 2y + z)(x + 2y - z)
$$