Definitions and notations

Quadratic Equation

A quadratic equation is a polynomial equation of second degree, which means the highest power of the variable is $2$. The standard form of a quadratic equation is given as:

$ax^2 + bx + c = 0$

where $a, b,$ and $c$ are constants, and $x$ is the variable. The coefficient $'a'$ cannot be equal to $0$ in a quadratic equation. The roots or solutions of a quadratic equation are the values of $x$ that satisfy the equation, i.e., the values of $x$ that make the equation equal to $0$.

Quadratic Formula

The quadratic formula is a formula that gives the solutions to the quadratic equation of the form $ax^2 + bx + c = 0$, where $a, b,$ and $c$ are constants, and $x$ is the variable. The quadratic formula is:

$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$

The quadratic formula provides the values of $x$ that satisfy the quadratic equation. The $±$ sign indicates that there are two possible solutions, which depend on whether the radical expression inside the formula is positive or negative. The quadratic formula can be derived by completing the square on the quadratic equation.

Roots of an Equation The roots of an equation are the values of the variables that satisfy the equation. For example, the roots of the equation $x^2 - 5x + 6 = 0$ are $x = 2$ and $x = 3$, since plugging in $2$ or $3$ for $x$ makes the equation true. In general, a polynomial equation of degree $n$ has $n$ roots, which may be real or complex numbers. The roots of an equation are also called its solutions.

Quartic or Bi-quadratic Equation

A quartic or biquadratic equation is a polynomial equation of degree $4$. The general form of a quartic equation is:

$ax^4 + bx^3 + cx^2 + dx + e = 0$

where $a, b, c, d,$ and $e$ are coefficients and $x$ is the variable. A quartic equation can have $0, 2,$ or $4$ real roots, and can also have complex roots.

A special case of the quartic equation is the biquadratic equation, where coefficients of variable with odd powers are zero so equation becomes: $ax^4 + bx^2 + c = 0$ Such equations are easily solvable by reducing to quadratic form

Exponential Equation An equation in which one or both sides contain exponential expressions. An exponential expression is one in which a variable appears in an exponent. Exponential equations can have one or more exponential terms and can be linear or nonlinear, depending on the power to which the variable is raised. For example, the equation $2^x = 16$ is an exponential equation with one exponential term.

Radical Equation A radical equation is an equation in which one or more variables appear inside a radical expression (such as square roots, cube roots, etc.). The goal of solving a radical equation is to isolate the variable under the radical sign and find its value. Some examples of radical equation are:
$\sqrt{x+2} - 4 = 0\\$$\\ \sqrt{5x-1} - \sqrt{x+3} = 1$