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Chapter 17: Problem 52

Solve the given problems. If \(x \neq y,\) show that \(x^{2}+y^{2}>2 x y\).

Short Answer

Expert verified
The inequality \(x^2 + y^2 > 2xy\) holds because \((x-y)^2 > 0\) when \(x \neq y\).

Step by step solution

01

Start with the given inequality

We need to show that \( x^2 + y^2 > 2xy \) when \( x eq y \). Let's rewrite the expression that we need to demonstrate: \( x^2 + y^2 - 2xy > 0 \).
02

Rearrange the expression

Notice that the expression \( x^2 + y^2 - 2xy \) can be rearranged into a perfect square: \((x - y)^2\). Therefore, we have \((x - y)^2 > 0\).
03

Analyze the expression

The rearranged expression \((x - y)^2\) is a perfect square. A perfect square \((u)^2\) is always greater than or equal to zero, and it is zero only when \(u = 0\).
04

Conclude inequality correctness

Since \(x eq y\), \((x - y)^2 > 0\) because \((x - y)\) is not zero. Thus, \(x^2 + y^2 > 2xy\) is indeed satisfied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square
In mathematics, a **perfect square** is an expression that is the square of a binomial.
In simpler terms, it takes the form of \((a imes a = a^2)\) or \((u)^2\), where \(u\) is some expression.
In our context, the perfect square involved is \((x - y)^2\). Here's why this is useful:
  • Perfect squares are beneficial in simplifying expressions and solving inequalities because their value is always non-negative.
  • Also, we've got the property that if a perfect square \((u)^2 = 0\), then \(u = 0\).

This property is crucial in this exercise because it helps conclude that since \((x - y)\) is not zero (given \(x eq y\)), \((x - y)^2\) is strictly greater than zero.
This beautifully demonstrates that \(x^2 + y^2 > 2xy\).
Rearrangement Technique
The **rearrangement technique** is a strategic method in algebra that allows us to rewrite expressions in a more manageable form.
In this problem, it involves transforming the expression \(x^2 + y^2 - 2xy\) into a perfect square.
This is achieved through the formula:
  • \((x - y)^2 = x^2 - 2xy + y^2\)

Using this identity, we can substitute \(x^2 + y^2 - 2xy\) with \((x - y)^2\), simplifying the inequality to \((x - y)^2 > 0\).
Simplifying through rearrangement like this not only helps identify perfect square structures but also reveals insightful properties of the expression.
This technique is a powerful tool in tackling both algebraic and inequality problems efficiently.
Mathematical Proof
A **mathematical proof** provides a complete and logically sound argument that a mathematical statement is true.
In our inequality problem, the proof involves a sequence of logical steps:
  • Start from the given statement \(x^2 + y^2 > 2xy\) which needs verification.
  • Convert it into \(x^2 + y^2 - 2xy > 0\).
  • Use rearrangement to recognize this as \((x - y)^2\).
  • Understand that since \(x eq y\), \((x - y)^2\) must be greater than zero.
  • Conclude by confirming \(x^2 + y^2 > 2xy\).

Proofs like this one help students develop a solid understanding of mathematical principles.
These exercises train logic and reinforce key concepts like algebraic manipulation and inequality testing.
Practically, mastering these proofs supports deeper learning and increases problem-solving skills efficiently.

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