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If x + 1/x = 5, then find the value of x┬▓ + 1/x┬▓.
23
25
27
21
23
For x+1/x=k, the value of x2+1/x2 is simply k2тИТ2. Here, 52тИТ2=25тИТ2=23.
The expression x + 1/x = 5 is provided.
(x+x1тАЛ)2=x2+x21тАЛ+2
For x+1/x=k, the value of x2+1/x2 is simply k2тИТ2. Here, 52тИТ2=25тИТ2=23.
Many students mistakenly square the expression as x2+1/x2=(x+1/x)2, forgetting to subtract the middle term 2(x)(1/x)=2.
Square both sides of the given equation
To obtain the x2 terms, square the given equation x+1/x=5 on both sides.
(x+x1тАЛ)2=52
Expand using algebraic identity
Expand the left side using the identity (a+b)2=a2+b2+2ab, where a=x and b=1/x.
x2+x21тАЛ+2(x)(x1тАЛ)=25
Simplify and solve
Since x cancels out in the middle term, we get x2+1/x2+2=25. Subtracting 2 from both sides yields the final result.
x2+x21тАЛ=25тИТ2=23
A is correct because squaring 5 and subtracting 2 gives the result 23.
This identity is a fundamental building block for higher-order power problems like x4+1/x4 or x3+1/x3, which frequently appear in competitive geometry and coordinate algebra questions.
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