Constructing quadratic function to prove the equal sign of Cauchy inequality Isn't the equal sign true when delta is equal to 0? Why let f (x) be equal to 0?

How can there be a delta if you don't let f (x) = 0

Prove the basic inequality, when to say if and only if the original inequality holds

It is usually necessary to judge whether it is necessary to emphasize this necessary and sufficient condition according to the things to be proved later. For example, it is necessary to ask a ^ 2 + B ^ 2 and 2Ab which is bigger. If only the former is greater than or equal to the latter, then logically, people who listen to the answer naturally have an extreme idea: "will it always be equal to". If you add a sentence if and only if blabla, Ordinary people don't go into it any more. When there is no need to talk about the necessary and sufficient conditions, the problem is usually clear at a glance, such as x > = 1, y

The inequality a / B + B / A is equal to or greater than 2 if

a=b≠0

Please use synthesis method and analysis method to prove inequality. 2 √ 2 - √ 7 ＜ 6 - √ 5 mathematics of senior two

[comprehensive method]
Method 1: (√ 6 - √ 5) = (√ 6 - √ 5) (√ 6 + √ 5) / (√ 6 + √ 5) = 1 / (√ 6 + √ 5)
In the same way
2√2-√7=√8-√7=1/(√8+√7)
√6+√51/(√8+√7)
So √ 6 - √ 5 > 2 √ 2 - √ 7
Method 2
（√6+√7）^2-(2√2+√5)^2
=13+2√42-(13+4√10)
=√4*42-√16*10
=√168-√160>0
[analytical method] to prove √ 6 - √ 5 ＞ 2 √ 2 - √ 7
Only need to prove (√ 6) + (√ 7) > (2 √ 2) + (√ 5)
Instant evidence √ 6 + √ 7 ＞ 8 + √ 5
As long as the syndrome (√ 6 ＋ 7) & # 178; > (2 √ 2 ＋ 5) & # 178;
13＋√42＞13＋√40
Instant evidence √ 42 ＞ 40
As long as certificate 42 > 40,
And 42 > 40 is obviously true,
So (√ 6) + (√ 7) > (2 √ 2) + (√ 5) holds, so √ 6 - √ 5 ＞ 2 √ 2 - √ 7

Constructing quadratic function to prove the equal sign of Cauchy inequality Isn't the equal sign true when delta is equal to 0? Why let f (x) be equal to 0?