Math Olympiad Square Root Simplification

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In this Math Olympiad Algebra Problem, you'll learn tips and tricks of solving International Math Olympiad exams quickly. #IMO #matholympiad #algebra #radicalequations #simplify #exponential

[@knotwilg3596]

Multiply numerator and denominator with the latter's complement Simplify by factoring out 4 Leaves V(6+2v5)) At this point most math savvies will "see" that 6 =5+1 and the solution is 1+v5 More formally, set the solution as a+bv5 where a and b are rationals; and a+bv5 must be positive Hence, a²+5b²=6 and ab= 1; Substitute b=1/a in the first equation and regroup a^4 - 6a²+ 5 =0, solve quadratic equation for A = a² a² = 1 or 5, so a is one of -1, 1, -v5, v5 and b = -1, 1, -1/v5, 1/v5 respectively The negative couples are invalid, because the sqrt must be positive And the remaining solutions 1+v5 and v5+1/v5*v5 are duplicates. This procedure applies to any problem of this form.

[@letsimage]

3:35 also u can just see without this formula, that there is a sum squared, because we have 2sqrt(5)

[@vultusalbus4216]

[Sqrt 5] + 1 equals 2φ

[@billv39]

I'm thinking that at 3:33 the term ab should be √(ab).

[@superacademy247]

I made a mistake

[@elmer6123]

Golden Ratio: Φ=(√5+1)/2 and 1/Φ=φ=(√5-1)/2; Φ^2=(√5+3)/2 and 1/Φ^2=φ^2=(3-√5)/2 Thus, (3-√5)=2φ^2 and 1/(3-√5)=1/(2φ^2)=Φ^2/2 and √[8/(3-√5)]=√(8Φ^2/2)=±2Φ=±(√5+1)

[@richardreiter6861]

Inside the formula you wrote inside the square-root should be 2 square root of ab ; not 2ab