If alpha and Beta are the zeros of the polynomial 2x2 – 3x + 5 then find.
1/a + 1/b +ab
a² + b²
here a is alpha and b is beta
Answers
ANSWER:
It looks something like this:
If you have heard of it before, then you should know that the steps are as follows:
1. Formation of an enolate using a base such as sodium hydroxide to steal a proton at the σ- carbon.
2. Nucleophilic attack of the σ,β− unsaturated ketone at the β− carbon, which is the common target for a nucleophile that is of the strength of a typical enolate.
3. The enolate takes a proton from the water that just formed from the hydroxide instead of attacking the carbonyl carbon because it cannot make any ring (a four-membered ring is too unstable).
4. The formation of another enolate occurs---this time, five carbons away from the leftmost carbonyl carbon, so a ring can be made (thus giving the sixth bond in a six-membered ring).
5. A ring is made via an intramolecular enolate nucleophilic attack.
6. This step is to show how the bond is connected. It is not actually realistically stable. :P
7. The water that just formed from the hydroxide that was used in step 4 donates a proton to the oxide intermediate.
8. Yet another enolate formation occurs. This time, facilitated by heat.
9. There is heat applied, so that, in addition to how the atoms are aligned, are such that the hydroxide can be eliminated from the product.
We have formed our product! :D As you can see, this is a ring formation reaction. It makes not one (like the acetylene synthesis), but TWO C−C bonds! NICE!
SIDENOTE: Steps 1−3 are called the Michael reaction. Steps 4−7 are an intramolecular aldol addition. Steps 8−9 form another σ,β− unsaturated ketone, from a β− hydroxyketone. Steps 4−9 are collectively called an intramolecular aldol condensation.
thnq