Solving equations

Most math skills boil down to being able to manipulate and solve equations. To solve an equation means to find the value of the unknown in the equation.

Check this shit out: \[ x^2-4=45. \]

To solve the above equation is to answer the question “What is $x$?” More precisely, we want to find the number which can take the place of $x$ in the equation so that the equality holds. In other words, we are asking \[ \text{"Which number times itself minus four gives 45?"} \]

That is quite a mouthful don't you think? To remedy this verbosity, mathematicians often use specialized mathematical symbols. The problem is that the specialized symbols used by mathematicians are confuse people. Sometimes even the simplest concepts are inaccessible if you don't know what the symbols mean.

What are your feelings about math, dear reader? Are you afraid of it? Do you have anxiety attacks because you think it will be too difficult for you? Chill! Relax my brothers and sisters. There is nothing to it. Nobody can magically guess what the solution is immediately. You have to break the problem down into simpler steps.

To find $x$, we can manipulate the original equation until we transform it to a different equation (as true as the first) that looks like this: \[ x= just \ some \ numbers. \]

That's what it means to solve. The equation is solved because you could type the numbers on the right hand side of the equation into a calculator and get the exact value of $x$.

To get $x$, all you have to do is make the right manipulations on the original equation to get it to the final form. The only requirement is that the manipulations you make transform one true equation into another true equation.

Before we continue our discussion, let us take the time to clarify what the equality symbol $=$ means. It means that all that is to the left of $=$ is equal to all that is to the right of $=$. To keep this equality statement true, you have to do everything that you want to do to the left side also to the right side.

In our example from earlier, the first simplifying step will be to add the number four to both sides of the equation: \[ x^2-4 +4 =45 +4, \] which simplifies to \[ x^2 =49. \] You must agree that the expression looks simpler now. How did I know to do this operation? I was trying to “undo” the effects of the operation $-4$. We undo an operation by applying its inverse. In the case where the operation is subtraction of some amount, the inverse operation is the addition of the same amount.

Now we are getting closer to our goal, namely to isolate $x$ on one side of the equation and have just numbers on the other side. What is the next step? Well if you know about functions and their inverses, then you would know that the inverse of $x^2$ ($x$ squared) is to take the square root $\sqrt{ }$ like this: \[ \sqrt{x^2} = \sqrt{49}. \] Notice that I applied the inverse operation on both sides of the equation. If we don't do the same thing on both sides we would be breaking the equality!

We are done now, since we have isolated $x$ with just numbers on the other side: \[ x = \pm 7. \]

What is up with the $\pm$ symbol? It means that both $x=7$ and $x=-7$ satisfy the above equation. Seven squared is 49, and so is $(-7)^2 = 49$ because two negatives cancel out.

If you feel comfortable with the notions of high school math and you could have solved the equation $x^2-4=25$ on your own, then you should consider skipping ahead to Chapter 2. If on the other hand you are wondering how the squiggle killed the power two, then this chapter is for you! In the next sections we will review all the essential concepts from high school math which you will need for the rest of the book. First let me tell you about the different kinds of numbers.