1.3 The roots of the equation \( f(x)=0 \) are \( x=\frac{4 \pm \sqrt{16-4 m(-m+5)}}{2 m} \) Determine the values of \( m \) for which the roots will be non-real.

To determine the values of \( m \) for which the roots of the equation \( f(x)=0 \) are non-real, we need to analyze the expression under the square root, known as the discriminant. The roots are given by: \[ x = \frac{4 \pm \sqrt{16 - 4m(-m+5)}}{2m} \] The roots will be non-real if the discriminant is less than zero. The discriminant in this case is: \[ D = 16 - 4m(-m + 5) = 16 + 4m^2 - 20m = 4m^2 - 20m + 16 \] We want to find the values of \( m \) for which: \[ 4m^2 - 20m + 16 < 0 \] Dividing the entire inequality by 4 gives us: \[ m^2 - 5m + 4 < 0 \] Now, we can factor the quadratic: \[ (m - 1)(m - 4) < 0 \] To find the intervals where this inequality holds, we can test values around the roots \( m = 1 \) and \( m = 4 \). The quadratic \( (m-1)(m-4) \) is negative between the roots: \[ 1 < m < 4 \] Thus, the values of \( m \) for which the roots of the equation are non-real are: \[ m \in (1, 4) \]