For what values of 'a' does the quadratic equation `x^(2) - ax + 1 = 0` not have real roots ?

Answer

To determine the values of 'a' for which the quadratic equation \( x^2 - ax + 1 = 0 \) does not have real roots, we need to analyze the discriminant of the equation. The discriminant (\( D \)) is given by the formula:

\[ D = b^2 - 4ac \]

For the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = 1 \)
- \( b = -a \)
- \( c = 1 \)

Substituting these values into the discriminant formula, we have:

\[ D = (-a)^2 - 4(1)(1) \]
\[ D = a^2 - 4 \]

For the quadratic equation to not have real roots, the discriminant must be less than zero:

\[ D < 0 \]

Substituting our expression for \( D \):

\[ a^2 - 4 < 0 \]

Now we solve this inequality:

1. Rearranging gives us:
\[ a^2 < 4 \]

2. Taking the square root of both sides, we find:
\[ -2 < a < 2 \]

Thus, the values of 'a' for which the quadratic equation \( x^2 - ax + 1 = 0 \) does not have real roots are:

\[ a \in (-2, 2) \]

### Summary of Steps:
1. Identify coefficients in the quadratic equation.
2. Write the discriminant formula.
3. Substitute coefficients into the discriminant.
4. Set the discriminant less than zero.
5. Solve the resulting inequality.