Quadratic Expressions

[Brandon W]

Quadratic Equations
G’day everyone. Welcome to my tutorial, this one I will be showing you how to find the value of “x” if you get a quadratic expression that is equal to zero. Also I will be showing you how to solve a quadratic expression, well part of the way hehe.
NOTE: This is not a programming tutorial! If you want you could interpret this to a script but if not this does not involve ANY programming!

Let’s get started;
First of all. The general form for a quadratic equation (also called a 2nd order polynomial) is:
Y = ax^2 +bx +c

Quadratic equations can be solved by factorising or by using the formula:


This will yield the two factors of the quadratic.

This might seem difficult but it isn’t as difficult as what you think. Take a look at this example.

Y = x^2 +2x +1
(x+1)(x+1)
x^2 +x +x +1
x^2 +2x + 1

That is the easy part. All we have done is turned the quadratic into a perfect square. Now this can be taken out by doing the following.

x * x
x * 1
1 * x
1 * 1

See the pattern? Now we can check this and turn it back into its original form. If it shows the same it is correct.

Now let’s apply the formula:


Remember the syntax of the quadratic? That is where we get a, b and c.
So let’s convert the formula using our quadratic.

(-2 +- sqrt(2^2 –(4*1*1)) / 2 x 1

Now we have replaced the a, b and c with our numbers. We can now work out the parts that we can. So let’s do that;

(-2 +- sqrt(4 –4)) / 2

Yet again, let’s go and work out what we can now.

(-2 +- sqrt(0)) / 2
-2 +- 0 / 2

I know most of use will be asking, what can we do now? The symbol (+-) indicates both sums need to be completed. So let’s do that now:

-2 + 0 /2

OR

-2 – 0 / 2

If you work out the two:

-2 / 2

OR

-2 / 2

That means that the answer to the quadratic:

x^2 +2x +1

Is -1 OR -1.


Here is a few more for you to work out, only if you want:

Y = x^2 +2x -3
Y = 2x^2 + 3x -4
Y = 6x^2 -5x +6

Now let’s move onto Part II. This one we will work out if the quadratic equal to 0.
“When the product of 2 expressions is equal to zero, then there are 2 possibilities for solving the equation. Either of the expression could equal zero.
For example:

(x + 4)(x – 5) = 0

Which then means that x = -4 or 5.

Let’s move onto a harder one.

(2x – 3)(2x + 15) = 0
Either (2x -3) = 0 or (2x + 15) = 0

You could write that down in an exam but it wouldn’t be sufficient so let’s go ahead a step and write this:

2x = 3
X = 3/2
X = 1.5

OR

2x = -15
X = -15 / 2
X = 7.5

Understand? I like to think that you just need to work out the parts of the equation that are equal to 0 and whenever there is a negative x is equal to a positive and vice versa. It happens only at a certain step though so be careful!


Not really a very detailed tutorial but I spent about an hour writing this tutorial so I hope use like it. I learnt it in Maths today so I thought I would write a tutorial. Correct me if I am wrong Winged hehe.

Not very useful but interesting to know. I wrote this because I LOVE maths 10x as much as Winged!

+rep if you found this helpful

[WingedPanther]

Something important to do is check the value of b^2-4ac. If it is negative, you will have 2 complex roots (which require special care in a programming language). If it is 0, you get a double root (-1 OR -1 above). If it is positive, you will have two roots.

[Xav]

Yum, quadratics... this formula always amazes me in how it always trumps up the two values for X.

[WingedPanther]

Xav, try performing the procedure for Completing the Squares on Ax^2 + Bx + C

[Xav]

I'm sorry? I don't get what you mean?

[WingedPanther]

Completing the Squares is a technique for solving quadratic equations. If you do it on the general equation, you get the quadratic formula.

[Brandon W]

Yer I know about that Winged. I have noticed it and it is really annoying to try and work with it.

[John]

Yum, quadratics... this formula always amazes me in how it always trumps up the two values for X.

bleh, after using the quadratic formula for for close to ten years now, it's no longer amazing, nor is the cubic formula or quartic formula (as interesting as that one may be). However what is truly amazing, is the fact that there is no quintic equation.

[Brandon W]

What are cubic and quartic formulas? I never heard of them before.

My maths teacher showed us how they come up with the quadratic formula, that is fairly big

[WingedPanther]

The cubic and quartic formulas are LONG and NASTY, especially the quartic. You can look them up on Google, but they won't provide much insight. I took a class that had proving the quintic is not generally solvable as one of its major conclusions.