List of Notes - Category: Quadratic Equations
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`.
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 can be derived from the steps for completing the square (actually, this formula is a general case).
Let's see how to do it.
- Start from the equation `ax^2+bx+c=0`.
- Divide both sides by `a`: `x^2+b/ax+c/a=0`.
- Move constant term to the right: `x^2+b/ax=-c/a`.
- 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)`.
- Rewrite left hand side: `(x+b/(2a))^2=-c/a+b^2/(4a^2)`.
- 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)`.
- Write the final equation: `(x+b/(2a))^2=(b^2-4ac)/(4a^2)`.
- Solve the equation: `x+b/(2a)=sqrt((b^2-4ac)/(4a^2))` or `x+b/(2a)=-sqrt((b^2-4ac)/(4a^2))`.
- Above equations have roots `x_1=(-b+sqrt(b^2-4ac))/(2a)` and `x_2=(-b-sqrt(b^2-4ac))/(2a)`.
- 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. 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.