Find a square number such that when twice its root is added to it or subtracted from it, one obtained other square numbers. In other words, solve a problem of the type. x^2+2x=u^2 x^2-2x=v^2

Find a square number such that when twice its root is added to it or subtracted from it, one obtained other square numbers. In other words, solve a problem of the type. x^2+2x=u^2 x^2-2x=v^2

Upper Level Math
asked 2021-03-07
Find a square number such that when twice its root is added to it or subtracted from it, one obtained other square numbers. In other words, solve a problem of the type.

Answers (1)

Step 1
To solve the given problem.
Step 2
Let us assume three squares such that
\(\displaystyle{a}^{{2}},{b}^{{2}}{\quad\text{and}\quad}{c}^{{2}}\) be three consecutive terms of an arithmetic progression with a common difference d.
put the value of \(\displaystyle{x}=\frac{{{2}{b}^{{2}}}}{{d}}\in\)...(i)
\(\displaystyle=\frac{{{4}{b}^{{2}}{c}^{{2}}}}{{d}^{{2}}}\) from(iii)
Step 3
Similarly Put the value of x in (ii)
\(\displaystyle=\frac{{{4}{b}^{{2}}{a}^{{2}}}}{{d}^{{2}}}\) from(iii)
Hence proved.

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