Category: Quadratic Equations

What is Quadratic Equation

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.

Incomplete Quadratic Equations

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`.

We already saw something similar in the incomplete quadratic equations note.

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.