Question

For the quadratic equation \[{x^2} - 2x + 1 = 0\], the value of \[x + \dfrac{1}{x}\] is:
A. -1
B. 1
C. 2
D. -2

Answer 1

- Hint: In this problem, we need to use the middle term splitter formula to find the factor of the given quadratic equation. We need to split the middle term in such a way that the sum is equal to the coefficient of \[x\] and the product is equal to the last term. Next, find the value of\[x\].

Complete step-by-step solution -
The given quadratic equation \[{x^2} - 2x + 1 = 0\] can be solved using the middle term splitting method as shown below.
\[
  \,\,\,\,\,\,{x^2} - 2x + 1 = 0 \\
   \Rightarrow {x^2} - \left( {1 + 1} \right)x + 1 = 0 \\
   \Rightarrow {x^2} - x - x + 1 = 0 \\
   \Rightarrow x\left( {x - 1} \right) - 1\left( {x - 1} \right) = 0 \\
\]
Further, solve the above equation.
\[
  \,\,\,\,\,\left( {x - 1} \right)\left( {x - 1} \right) = 0 \\
   \Rightarrow {\left( {x - 1} \right)^2} = 0 \\
   \Rightarrow x = 1,1 \\
\]
Now, substitute 1 for \[x\] in the expression \[x + \dfrac{1}{x}\] to obtain the value of it.
\[
  \,\,\,\,\,\,x + \dfrac{1}{x} \\
   \Rightarrow 1 + \dfrac{1}{1} \\
   \Rightarrow 1 + 1 \\
   \Rightarrow 2 \\
\]
Thus, the value of the expression \[x + \dfrac{1}{x}\] is 2, hence, option (C) is the correct answer.

Note: While doing the factor of a quadratic equation of the form \[a{x^2} + bx + c = 0\] using middle term split method, we need to split the middle term in such a way that the sum is equal to \[b\] and product is equal to\[c\].