# Inequality Proof

## Homework Statement

Prove that $$\left(ab+cd\right)^{2} \leq \left(a^{2}+c^{2}\right)\left(b^{2}+d^{2}\right)$$

None

## The Attempt at a Solution

I've broken the LHS down to the following:

$$\left(ab\right)^{2}+2abcd+\left(cd\right)^{2}$$

The RHS:

$$\left(ab\right)^{2} + \left(ad\right)^{2} + \left(bc\right)^{2} + \left(cd\right)^{2}$$

So, ultimately... it works out that I need to show $$2abcd \leq \left(ad\right)^{2} + \left(bc\right)^{2}$$

This is where I'm getting stuck... Any suggestions...

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## Homework Statement

Prove that $$\left(ab+cd\right)^{2} \leq \left(a^{2}+c^{2}\right)\left(b^{2}+d^{2}\right)$$

None

## The Attempt at a Solution

I've broken the LHS down to the following:

$$\left(ab\right)^{2}+2abcd+\left(cd\right)^{2}$$

The RHS:

$$\left(ab\right)^{2} + \left(ad\right)^{2} + \left(bc\right)^{2} + \left(cd\right)^{2}$$

So, ultimately... it works out that I need to show $$2abcd \leq \left(ad\right)^{2} + \left(bc\right)^{2}$$

This is where I'm getting stuck... Any suggestions...
Yup, so far so good, now subtract 2abcd from both sides, and you'll get:

$$(ad) ^ 2 - 2(ad)(bc) + (bc) ^ 2 \geq 0$$

Does the LHS of this inequality remind you of something? :surprised

Yup, so far so good, now subtract 2abcd from both sides, and you'll get:

$$(ad) ^ 2 - 2(ad)(bc) + (bc) ^ 2 \geq 0$$

Does the LHS of this inequality remind you of something? :surprised

Wow, long night...
http://scienceblogs.com/insolence/facepalm.jpg [Broken]​
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Last edited by a moderator:
Just use (A-B)2=A2-2AB+B2.

Regards.

## Homework Statement

Prove that $$\left(ab+cd\right)^{2} \leq \left(a^{2}+c^{2}\right)\left(b^{2}+d^{2}\right)$$

None

## The Attempt at a Solution

I've broken the LHS down to the following:

$$\left(ab\right)^{2}+2abcd+\left(cd\right)^{2}$$

The RHS:

$$\left(ab\right)^{2} + \left(ad\right)^{2} + \left(bc\right)^{2} + \left(cd\right)^{2}$$

So, ultimately... it works out that I need to show $$2abcd \leq \left(ad\right)^{2} + \left(bc\right)^{2}$$

This is where I'm getting stuck... Any suggestions...