Question

The expression \(x^2-x+1=0\) has

• A. One proper linear factor
• B. Two proper linear factors
• C. No proper linear factor
• D. Cannot be determined

Hint:

A quadratic has real linear factors only when: \[ D=b^2-4ac\geq0 \]

Answer 1

The Correct Option is C

Concept:
A quadratic equation: \[ ax^2+bx+c=0 \] has real linear factors if its discriminant is non-negative. The discriminant is: \[ D=b^2-4ac \] If: \[ D>0 \] then two distinct real linear factors exist. If: \[ D=0 \] then one repeated real linear factor exists. If: \[ D<0 \] then there is no real linear factor.

Step 1: Identify coefficients.
Given: \[ x^2-x+1=0 \] Compare with: \[ ax^2+bx+c=0 \] So: \[ a=1,\quad b=-1,\quad c=1 \]

Step 2: Calculate discriminant.
\[ D=b^2-4ac \] \[ D=(-1)^2-4(1)(1) \] \[ D=1-4 \] \[ D=-3 \]

Step 3: Interpret discriminant.
Since: \[ D=-3<0 \] the quadratic has no real roots. Therefore, it cannot be factorized into proper real linear factors.

Step 4: Check the options.
Option (A) one proper linear factor is incorrect.
Option (B) two proper linear factors is incorrect.
Option (C) no proper linear factor is correct.
Option (D) cannot be determined is incorrect because discriminant clearly decides it. Hence, the correct answer is: \[ \boxed{(C)\ \text{No proper linear factor}} \]