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If x+x1=6, then x2+x21=?
38
36
34
32
34
For an equation x+x1=k, the result for x2+x21 is simply k2−2. In this case, 62−2=36−2=34.
Given that x + 1/x = 6.
(a+b)2=a2+2ab+b2
For an equation x+x1=k, the result for x2+x21 is simply k2−2. In this case, 62−2=36−2=34.
Many students mistakenly calculate (x+x1)2 as x2+x21 and forget to subtract the middle term 2(x)(x1)=2.
Square both sides
To isolate x2+x21, we square both sides of the given equation x+x1=6.
(x+x1)2=62
Expand the equation
Using the algebraic identity (a+b)2=a2+2ab+b2, we expand the left side where a=x and b=x1.
x2+2(x)(x1)+x21=36
Simplify and solve
Since x⋅x1=1, the equation becomes x2+2+x21=36. Subtracting 2 from both sides gives the result.
x2+x21=36−2=34
C is correct because squaring the given expression results in 34.
This shortcut k2−2 is a fundamental property used in various competitive exams for problems involving higher powers like x4+x41 by applying the square rule again.
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