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Question

If x=462+3. find the value of x+22x22+x+23x23, using the properties of proportion.

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Solution

Given, x=462+3
It may be written as,
x=22×232+3
or x22=232+3(i)
By componendo and dividendo, we get
x+22x22=23+2+32323=33+232(ii)
Equation (i) may also be written as
x23=222+3
By componendo and dividendo, we get
x+23x23=22+2+32223=32+323(iii)
Adding equation (ii) and (iii), we get
x+22x22+x+23x23=33+232+32+323=33+232(32+332)=33+232332=232232=2(32)32=2

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