Quadratic Equation (ax^2+bx+c=0) is a special case of the quadratic function (y=ax^2+bx+c) when the function value is equal to zero. For certain problems involving quadratic functions, Vieta's theorem can be used to relate the roots of the quadratic equation to its coefficients. The distribution of the roots can also be determined intuitively by examining the graph of the quadratic function. The points where the graph intersects the x-axis and the position of the graph can also be determined using the discriminant.

The formula for the discriminant is derived as follows:

[y = ax^2 + bx + c]

Rewriting it in vertex form:

[y = a(x + \frac{b}{2a})^2 + \frac{4ac-b^2}{4a}]

Let's solve for (y=0):

[a(x + \frac{b}{2a})^2 + \frac{4ac-b^2}{4a} = 0]

Clearing the denominator:

[4a^2(x + \frac{b}{2a})^2 + 4ac-b^2 = 0] [4a^2(x + \frac{b}{2a})^2 = b^2 - 4ac]

The left side of the equation is a non-negative number, so:

• When (b^2 - 4ac < 0), there are no real solutions.
• When (b^2 - 4ac = 0), there is one real solution.
• When (b^2 - 4ac > 0), there are two distinct real solutions.