Quadratic equation in one variable is the equation with standard form color(purple)(ax^2+bx+c=0).

a, b and c are some numbers and x is variable. Note, that a can't be zero.

In essence, quadratic equation is nothing more than quadratic polynomial ("quad" means square) on the left hand side, and zero on the right hand side.

Quadratic equation ax^2+bx+c=0 is called incomplete, if either b or c (or both) equals 0.

Such equations can be easily solved without advanced methods.

Example 1. Solve x^2-81=0.

Here a=1, b=0, c=-81.

## Solving Quadratic Equations by Completing the Square

Completing the square is a method for solving quadratic equation.

Before we into the method itself, let's start from a simple example.

Example 1. Solve equation (x-3)^2=16.

## Quadratic Equation Formula and the Discriminant

Quadratic Equation Formula can be derived from the steps for completing the square (actually, this formula is a general case).

Let's see how to do it.

1. Start from the equation ax^2+bx+c=0.
2. Divide both sides by a: x^2+b/ax+c/a=0.
3. Move constant term to the right: x^2+b/ax=-c/a.
4. Add (b/2a)^2=b^2/(4a^2) to both sides of the equation: x^2+b/ax+b^2/(4a^2)=-c/a+b^2/(4a^2).
5. Rewrite left hand side: (x+b/(2a))^2=-c/a+b^2/(4a^2).
6. Simplify right hand side: -c/a+b^2/(4a^2)=(-c*color(red)(4a))/(a*color(red)(4a))+b^2/(4a^2)=(-4ac)/(4a^2)+b^2/(4a^2)=(b^2-4ac)/(4a^2).
7. Write the final equation: (x+b/(2a))^2=(b^2-4ac)/(4a^2).
8. Solve the equation: x+b/(2a)=sqrt((b^2-4ac)/(4a^2)) or x+b/(2a)=-sqrt((b^2-4ac)/(4a^2)).
9. Above equations have roots x_1=(-b+sqrt(b^2-4ac))/(2a) and x_2=(-b-sqrt(b^2-4ac))/(2a).
10. We can write it even more compactly: x_(1,2)=(-b+-sqrt(b^2-4ac))/(2a).

Expression b^2-4ac is called the discriminant of the quadratic equation.

## Viet Theorem

Viet Theorem. If quadratic equation ax^2+bx+c=0 (reduced form is x^2+b/a+c/a=0) has roots p and q, then color(green)(p+q=-b/a), color(magenta)(pq=c/a), i.e. sum of the roots equals second coefficient, taken with opposite sign, and product of roots equals constant.