Asked by Hana
If a=root 3-root 2/root3+root2 and b=root3+root2/root3-root 2 ,then find a square+b square-5ab
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Answered by
GPT-5 mini
AI
a = (√3 − √2)/(√3 + √2), b = (√3 + √2)/(√3 − √2).
Note ab = 1. Also
a + b = (√3 − √2)/(√3 + √2) + (√3 + √2)/(√3 − √2)
= ((√3 − √2)^2 + (√3 + √2)^2)/((√3)^2 − (√2)^2)
= (5 − 2√6 + 5 + 2√6)/(3 − 2) = 10.
Hence a^2 + b^2 = (a + b)^2 − 2ab = 100 − 2 = 98.
Therefore a^2 + b^2 − 5ab = 98 − 5·1 = 93.
Note ab = 1. Also
a + b = (√3 − √2)/(√3 + √2) + (√3 + √2)/(√3 − √2)
= ((√3 − √2)^2 + (√3 + √2)^2)/((√3)^2 − (√2)^2)
= (5 − 2√6 + 5 + 2√6)/(3 − 2) = 10.
Hence a^2 + b^2 = (a + b)^2 − 2ab = 100 − 2 = 98.
Therefore a^2 + b^2 − 5ab = 98 − 5·1 = 93.
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