Prove $(a + b)^2 \geq 4ab$ 
Prove $(a + b)^2 \geq 4ab$

What direction should I take with this proof? Can I use induction here? or is there a better method?
I tried a few manipulations, but couldn't seem to find a form that proved it for all $x$.
One such manipulation resulted in:
$a^2 + b^2 \geq 2ab$
which seems close to the triangle inequality. Can I use this somehow?
 A: Here is an alternative proof, which is circuitous but perhaps enjoyable as a curiosity:
Fact: For any fixed $P > 0$, the rectangle of perimeter $P$ with maximal area is a square.

Consequence: Given a $2a\times 2b$ rectangle, it has area $4ab$ and perimeter $4a + 4b$. But the square of the same perimeter has side-length $a+b$, hence area $(a+b)^2$.
The Fact above now provides the desired inequality in terms of square and rectangle areas:
$$(a+b)^2 \geq 4ab $$

Fact Proof (for completeness): Let us denote one side of the rectangle by $x$ and the other by $y$. Then the perimeter is $P = 2x + 2y$, so that $2y = P - 2x$, whence $y = P/2 - x$.
Now the area is $xy = x(P/2 - x) = Px/2 - x^2$. To maximize the last expression, observe that it is a downward facing parabola; so its maximum occurs at the vertex, i.e., $P/4$. 
Thus, $x = P/4 = y$ as claimed.
A: You are on the right track; but try subtracting $4ab$ from both sides of the original inequality, instead of just $2ab$. Do you recognize what you get on the left? Finally, remember that $x^2\geq0$ for any real number $x$.
A: You're pretty close. Set it up as a series of if and only if statements.
$$
(a+b)^2\geq 4ab \Leftrightarrow a^2+2ab+b^2\geq 4ab \Leftrightarrow a^2-2ab+b^2\geq 0 \Leftrightarrow (a-b)^2\geq 0
$$
and the last statement is clearly always true.
A: short proof
$$
(a+b)^2=4ab+(a-b)^2
$$
A: First note that $\forall a,b\in\mathbb{R}$ $$(a-b)^{2}\ge 0.$$ $$\Rightarrow$$ $$a^{2}+b^{2}\ge2ab$$$$\Rightarrow$$
$$a^{2}+b^{2}+2ab\ge 4ab$$ $$\Rightarrow$$ $$(a+b)^{2}\ge 4ab.$$ So, a good method for solving these types of problems is to assume that the inequality is true and see where it brings you, then work backwards. Let me know if you need any clarifications!
