Question

if x= √3-√2/√3+√2 and y= √3+√2/√3-√2, then x² +xy + y² ​

Answer 1

Answer:

x=√3-√2/√3+√2 & y=√3+√2/√3-√2.

x²+xy+y²=?.

x=√3-√2/√3+√2.

Rationalising denominator of √3+√2 is √3-√2.

Multiply numerator & denominator by √3-√2.

x=√3-√2/√3+√2×√3-√2/√3-√2.

x=(√3-√2)²/(√3)²-(√2)². (Because (a+b)×(a-b)=a²+b² where a=√3 & b=√2).

x=(√3)²+(√2)²-2(√3)(√2)/3-2. (Because (a-b)²=a²+b2-2ab where a=√3 & b=√2).

x=3+2-2√6/1. (Because √3×√2=√6).

x=5-2√6.

y=√3+√2/√3-√2.

Rationalising denominator of √3-√2 is √3+√2.

Multiply numerator & denominator by √3+√2.

y=√3+√2/√3-√2×√3+√2/√3+√2.

y=(√3+√2)²/(√3)²-(√2)². (Because (a-b)×(a+b)=a²+b² where a=√3 & b=√2).

y=(√3)²+(√2)²+2(√3)(√2)/3-2. (Because (a+b)²=a²+b2+2ab where a=√3 & b=√2).

y=3+2+2√6/1. (Because √3×√2=√6).

y=5+2√6.

x²+xy+y²=(5-2√6)²+(5-2√6)(5+2√6)+(5+2√6)². (Because x=5-2√6 & y=5+2√6).

x²+xy+y²=((5)²+(2√6)²-2(5)(2√6))+((5)²-(2√6)²)+((5)²+(2√6)²+2(5)(2√6)). (Because (a-b)²=a²+b²-2ab, (a-b)(a+b)=a²-b² & (a+b)²=a²+b²+2ab where a=5 & b=2√6).

x²+xy+y²=(25+4×6-20√6)+(25-4×6)+(25+4×6+20√6).

x²+xy+y²=(25+24-20√6)+(25-24)+(25+24+20√6).

x²+xy+y²=25+24-20√6+25-24+25+24+20√6. (Because +×-=-).

x²+xy+y²=49-20√6+1+49+20√6. (Because +×-=- & largest number is important).

x²+xy+y²=49+1+49.

x²+xy+y²=99.

I think this is your answer.