Solving quadratic functions

A Quick Review:

When you solve or find the roots of quadratic functions you are finding its x-intercepts. A quadratic function can have 0, 1, or 2 roots.

The Four Methods:

  • Factoring
  • Graphing
  • The Quadratic Formula
  • Completing the Square

Each of these methods has many upsides and downsides. A teacher or a tutor can suggest a method to a student, but ultimately it is up to the student to decide which method they are the most comfortable with using.



  • Quick
  • Neat/Not Messy
  • Typically Well Practiced (I mean like you do spend a couple months learning this)


  • Doesn’t always work


2x²-x-1=0 –> (2x+1)(x-1)

(2x+1)=0 and (x-1)=0

2x=-1 and x=1

x= -½ and 1

(I’m sorry this article won’t teach how to do each method, it will only show a brief example of each)



  • very easy using a calculator
  • calculator will give you correct answer if you put function in correctly (very little room for human error if done on calculator)


  • not the easiest to do on paper


Screen Shot 2017-06-17 at 12.47.47 AM.png

The roots of this equation are -1 and 1.

The Quadratic Formula:


  • It always works!


  • If you don’t memorize it 100% correctly it won’t work.



x=(-(2)±√((2)²-4(4)(-3)) /2(4)

x=(-2±√52)  /8

x=(-2±2√13) /8

x=-1±√13) /4

Completing the Square:


  • Always works


  • Can get messy








What I do:

I always try to factor it first because it is the easiest and quickest method. If I can’t factor it then I use the quadratic formula because it is the most straightforward.

Where to start:

I’d always use the quadratic formula if I was confused because it works every single time. Once I got the hang of solving quadratic equations, I’d explore the other methods as much as I can. Mostly time and experience will help you figure out the best way for you to solve quadratic equations. Please remember that everyone is different and just because one way works the best for me, you, or your friend doesn’t mean it’ll work the best for somebody else.


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