91Ó°ÊÓ

A quadratic equation is a type of polynomial equation in which the highest exponent of the variable is 2. This means the general form of a quadratic equation is given as: ax^2 + bx + c = 0.
In this equation:

• 'a' is the coefficient of x^2 and should not be zero, as that would make the equation linear, not quadratic.
• 'b' is the coefficient of x.
• 'c' is the constant term.

In the given problem, our quadratic equation is: x^2 - 4x - 1 = 0, where:

• 'a' is 1,
• 'b' is -4,
• 'c' is -1.

To solve this kind of equation, we use methods like factoring, completing the square, graphing, or the quadratic formula. In this exercise, we will use the quadratic formula.

The discriminant is a key part of the quadratic formula, and it can tell us a lot about the nature of the solutions of a quadratic equation. The discriminant is found inside the square root of the quadratic formula, and it is calculated as:
b^2 - 4ac.
The value of the discriminant helps us determine:

• If it is positive (>0), there are two distinct real solutions.
• If it is zero, there is exactly one real solution (known as a repeated or double root).
• If it is negative (<0), there are no real solutions (the solutions will be complex or imaginary).

For our equation, we compute the discriminant as follows:

(-4)^2 - 4(1)(-1)
= 16 + 4
= 20.
This result is positive, meaning our quadratic equation has two distinct real solutions.

When solving a quadratic equation, our goal is to find the values of x that make the equation true. These values are known as the 'solutions' or 'roots' of the equation. Real solutions are the values of x that can be plotted on the number line, and they come from the real number system.
The quadratic formula: x = \[ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] is used to find these solutions. Here, the '±' sign indicates that there are generally two solutions, depending on the value of the discriminant.
In our specific problem, we substitute the known values into the quadratic formula: x = \[ \frac{4 \pm \sqrt{20}}{2} \] = \[ \frac{4 \pm 2 \sqrt{5}}{2} \] = 2 ± \( \sqrt{5} \).
Thus, the real solutions to the equation x^2 - 4x - 1 = 0 are:

• x = 2 + \( \sqrt{5} \), and


• x = 2 - \( \sqrt{5} \).

These values represent the two distinct points where the parabola intersects the x-axis.