Quadratic Forms

A quadratic form is a polynomial of degree two in multiple variables, typically expressed as $$Q(x_1, x_2, \text{...}, x_n) = a_1x_1^2 + a_2x_2^2 + ... + a_nx_n^2 + b_1x_1x_2 + ... + b_{n-1}x_{n-1}x_n + c$$ .

The standard form of a quadratic equation is $$ax^2 + bx + c = 0$$ , where $$a$$ , $$b$$ , and $$c$$ are constants and @@a eq 0@@.

The vertex of a quadratic function in the form $$f(x) = ax^2 + bx + c$$ can be found using the formula $$x = -\frac{b}{2a}$$ . The y-coordinate can be found by substituting this x-value back into the function.

The axis of symmetry of a quadratic function $$f(x) = ax^2 + bx + c$$ is the vertical line given by the equation $$x = -\frac{b}{2a}$$ .

The roots of a quadratic equation are the values of $$x$$ that satisfy the equation $$ax^2 + bx + c = 0$$ . They can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ .

The discriminant, given by $$D = b^2 - 4ac$$ , indicates the nature of the roots of the quadratic equation: if $$D > 0$$ , there are two distinct real roots; if $$D = 0$$ , there is one real root (a repeated root); if $$D < 0$$ , there are no real roots.

The vertex form of a quadratic function is given by $$f(x) = a(x - h)^2 + k$$ , where $$(h, k)$$ is the vertex of the parabola.

To convert from standard form $$f(x) = ax^2 + bx + c$$ to vertex form, complete the square: factor out $$a$$ from the first two terms, then adjust the constant term accordingly.

The leading coefficient $$a$$ in the quadratic function $$f(x) = ax^2 + bx + c$$ determines the direction of the parabola: if $$a > 0$$ , the parabola opens upwards; if $$a < 0$$ , it opens downwards.

For a quadratic equation $$ax^2 + bx + c = 0$$ , the sum of the roots is given by $$-\frac{b}{a}$$ and the product of the roots is given by $$\frac{c}{a}$$ .