if a+b+√6=√3+√2/√3-√2 find a and b
Answers
ANSWER
This problem can be solved in 2 ways.
1st :-
a=3–√+2–√3–√−2–√a=3+23−2
=(3–√+2–√)(3–√+2–√)(3–√−2–√)(3–√+2–√)=(3+2)(3+2)(3−2)(3+2)
=(3–√+2–√)23−2=(3+2)23−2
=((3–√)2+(2–√)2+2(3–√)(2–√)=((3)2+(2)2+2(3)(2)
=3+2+26–√=3+2+26
=5+26–√=5+26
b=3–√−2–√3–√+2–√b=3−23+2
=(3–√−2–√)(3–√−2–√)(3–√+2–√)(3–√−2–√)=(3−2)(3−2)(3+2)(3−2)
=(3–√−2–√)23−2=(3−2)23−2
=((3–√)2+(2–√)2+−2(3–√)(2–√)=((3)2+(2)2+−2(3)(2)
=3+2−26–√=3+2−26
=5−26–√=5−26
Now
a2+b2a2+b2
We know that
(x+y)2=x2+y2+2xy(x+y)2=x2+y2+2xy
⟹x2+y2=(x+y)2−2xy⟹x2+y2=(x+y)2−2xy
If we take x=ax=a and y=by=b then
a2+b2a2+b2
=(a+b)2−2ab=(a+b)2−2ab
=(5+26–√+5−26–√)2−2((5+26–√)(5−26–√))=(5+26+5−26)2−2((5+26)(5−26))
=(10)2−2(25−24)=(10)2−2(25−24)
Here we use identity (a+b)(a−b)=a2−b2(a+b)(a−b)=a2−b2
=100−2=100−2
=98=98
2nd :-
a=3–√+2–√3–√−2–√a=3+23−2
b=3–√−2–√3–√+2–√b=3−23+2
Now
ab=3–√+2–√3–√−2–√×3–√−2–√3–√+2–√ab=3+23−2×3−23+2
=1=1
a+b=3–√+2–√3–√−2–√+3–√−2–√3–√+2–√a+b=3+23−2+3−23+2
=(3–√+2–√)2+(3–√−2–√)2(3–√+2–√)(3–√−2–√)=(3+2)2+(3−2)2(3+2)(3−2)
=5+26–√+5−26–√3−2=5+26+5−263−2
=10=10
Now
(x+y)2=x2+y2+2xy(x+y)2=x2+y2+2xy
⟹x2+y2=(x+y)2−2xy⟹x2+y2=(x+y)2−2xy
Replacing x=a,y=bx=a,y=b
a2+b2=(a+b)2−2aba2+b2=(a+b)2−2ab
=(10)2−2×1=(10)2−2×1
Replacing (x+y)=10(x+y)=10 and xy=1xy=1
=100−2=100−2
=98=98
Therefore a2+b2=98a2+b2=98
It totally depends on you which method to use but I recommend the second one .
Step-by-step explanation: