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Examoogle • User • info@examoogle.com • EE-2024-8821
Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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If x + 1/x = 4, then x² + 1/x² is:
12
14
16
18
14
For the equation x+x1=k, the value of x2+x21 is always k2−2. Here, 42−2=16−2=14.
The expression x + 1/x = 4 is given.
(x+x1)2=x2+x21+2
For the equation x+x1=k, the value of x2+x21 is always k2−2. Here, 42−2=16−2=14.
Many students mistakenly square the expression x+x1 and forget to subtract 2, resulting in the incorrect answer 16.
Square both sides
To obtain terms with squares, square the given equation (x+x1)=4 on both sides.
(x+x1)2=42
Expand using (a+b)2
Expand the left side using the algebraic identity (a+b)2=a2+2ab+b2, where a=x and b=1/x.
x2+2(x)(x1)+x21=16
Simplify and solve
The term 2(x)(1/x) simplifies to 2. Subtracting 2 from both sides gives the value of x2+1/x2.
x2+2+x21=16⟹x2+x21=14
B is correct because the application of the identity (x+1/x)2=x2+1/x2+2 yields 42−2=14.
This shortcut k2−2 is extremely useful in Polynomials and Quadratic equations chapters for finding higher power sums like x4+1/x4 by repeating the process.
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