Proving Quadratic Inequality: (x-y)^2 ≥ 0

In summary, by expanding (x-y)^2 and adding 2xy to both sides of the trivial inequality, we can prove that x^2 + y^2 ≥ 2xy for all real numbers x and y. This is related to the AM-GM inequality in the two variable case.
  • #1
sg001
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Homework Statement



By expanding (x-y)^2, prove that x^2 +y^2 ≥ 2xy for all real numbers x & y.


Homework Equations





The Attempt at a Solution


expanding (x-y)^2

x^2 - 2xy + y^2= 0

Hence, x^2 + y^2 = 2xy

But where does the ≥ come into it? and why?
when you put values in (except i which is not real of course) they all come out as = 2xy, which does satisfy ≥2xy, but why does this come into it??
Some insight would be fantastic!
 
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  • #2
If x and y are real then (x-y) is real. (x-y)^2>=0, yes?
 
  • #3
Ohhhh I see...
Thanks!
 
  • #4
Continuing from Dick's hint:

(x-y)^2 ≥ 0 (Trivial inequality)
x^2-2xy+y^2 ≥ 0
And adding 2xy to both sides, we get:
x^2+y^2 ≥ 2xy as desired.

And if you're up for it, try proving the two variable case of the AM-GM inequality from here (it's pretty simple).
 

FAQ: Proving Quadratic Inequality: (x-y)^2 ≥ 0

What is a quadratic inequality?

A quadratic inequality is an inequality that contains a quadratic expression, which is an expression with a variable raised to the second power. It can be written in the form ax^2 + bx + c < 0 (or > 0) where a, b, and c are constants and x is a variable.

How do you solve a quadratic inequality?

To solve a quadratic inequality, you must first factor the quadratic expression and then use the signs of the factors to determine the intervals that make the inequality true. The solution will be the union of these intervals. You can also graph the quadratic inequality on a number line to visualize the solution.

What is the difference between a quadratic equation and a quadratic inequality?

A quadratic equation is an equation that is set equal to zero and can be solved to find the value(s) of the variable. A quadratic inequality, on the other hand, is an inequality that compares two expressions and can have multiple solutions or intervals. The solution to a quadratic inequality is a set of values that make the inequality true, rather than a specific value.

How do you prove a quadratic inequality?

To prove a quadratic inequality, you can use algebraic manipulation, the properties of inequalities, and the properties of quadratic equations. You may also need to use logic and reasoning to explain your steps and show that the inequality holds true for all values of the variable.

What are some real-world applications of quadratic inequalities?

Quadratic inequalities can be used to model and solve various real-world problems, such as maximizing profits for a business, determining the range of possible values for measurements, and finding the optimal dimensions of an object. They can also be used in physics to describe the motion of objects under the influence of gravity or air resistance.

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