Question

# Write the general form of a quadratic polynomial.

Hint- Here, we will proceed by defining the quadratic polynomial and then writing down the general form of any quadratic polynomial in three variables (x, y and z), in two variables (x and y) and in one variable (x).

A quadratic polynomial is a polynomial of degree 2 or simply we can say that a quadratic polynomial is a polynomial function with one or more variables in which the highest degree term is of the second degree.
The general form of any quadratic polynomial in three variables i.e., x, y and z is given as under
$f\left( {x,y,z} \right) = a{x^2} + b{y^2} + c{z^2} + dxy + eyz + fxz + gx + hy + iz + j$ where a, b, c, d, e, f, g, h, i, j are all constants.
The general form of any quadratic polynomial in two variables i.e., x and y is given as under
$f\left( {x,y} \right) = a{x^2} + b{y^2} + cxy + dx + ey + f$ where a, b, c, d, e, f are all constants.
The general form of any quadratic polynomial in one variable i.e., x is given as under
$f\left( x \right) = a{x^2} + bx + c$ where a, b, c are all constants.
When these polynomials are equated with zero, then that equation is termed as a quadratic equation.
The general form of any quadratic equation in three variables i.e., x, y and z is given as under
$f\left( {x,y,z} \right) = a{x^2} + b{y^2} + c{z^2} + dxy + eyz + fxz + gx + hy + iz + j = 0$ where a, b, c, d, e, f, g, h, i, j are all constants.
The general form of any quadratic equation in two variables i.e., x and y is given as under
$f\left( {x,y} \right) = a{x^2} + b{y^2} + cxy + dx + ey + f = 0$ where a, b, c, d, e, f are all constants.
The general form of any quadratic equation in one variable i.e., x is given as under
$f\left( x \right) = a{x^2} + bx + c = 0$ where a, b, c are all constants.

Note- In this particular problem, out of all the quadratic polynomials the general form of quadratic polynomial in one variable is usually used. The solution of the quadratic equation in one variable i.e., $f\left( x \right) = a{x^2} + bx + c = 0$ is either solved by factorization method or by discriminant method.