- Identify the coefficients \(a=1\), \(b=-4\), and \(c=5\).
- Calculate the discriminant using \(\Delta = b^2 - 4ac = (-4)^2 - 4(1)(5) = -4\).
- Determine the nature of the roots by observing that \(\Delta < 0\), implying two complex solutions.
- Conclude that the correct options are A and C.
- Final Answer: \(\boxed{A, C}\)
Explanation
- Identifying the coefficients of the quadratic equation
To determine the discriminant and the nature of the solutions for the quadratic equation \(x^2 - 4x + 5 = 0\), we first identify the coefficients of the quadratic expression. A standard quadratic equation is given by \(ax^2 + bx + c = 0\). By comparing this to our given equation, we find that \(a = 1\), \(b = -4\), and \(c = 5\).
- Calculating the discriminant
The discriminant \(\Delta\) of a quadratic equation is defined by the formula \(\Delta = b^2 - 4ac\). Using the coefficients identified, we calculate:
\[\Delta = (-4)^2 - 4(1)(5) = 16 - 20 = -4.\]
Thus, the value of the discriminant is \(-4\).
- Determining the nature of the solutions
The nature of the solutions is determined by the sign of the discriminant \(\Delta\):
- If \(\Delta > 0\), there are two distinct real solutions.
- If \(\Delta = 0\), there is exactly one real solution (a repeated root).
- If \(\Delta < 0\), there are two complex (non-real) solutions.
Since we calculated \(\Delta = -4\), which is less than 0, the equation has two complex solutions.
- Final conclusion
Based on our analysis, the discriminant is \(-4\) (Option C) and the equation has two complex solutions (Option A). Thus, the correct choices are A and C.
Examples
The discriminant is a vital tool in engineering and physics, helping to determine the stability of systems. For example, when analyzing the motion of a damped pendulum, the nature of the roots of its characteristic equation (which is quadratic) tells us whether the system will oscillate or return directly to equilibrium without oscillating. Calculating the discriminant allows engineers to classify these behaviors quickly and effectively.