Determining the value of x in a Quadratic Equation?

Question

My question isn't specific to a particular quadratic problem but rather it is applicable to all quadratic equations that can be solved through factorization.

So far I have understood most of the steps involved in solving a quadratic equation except one the last step -- after factoring and simplifying I get an answer $(x+3)(x+2)=0$ and immediately after this the next step involves writing the value of x as either 3 or 2. I'm sure there's an extra step in between these two that most teachers skip out on. Why does $(x+3)(x+2)=0$ indicate that $x = 3$ or $2$?

Math is not at all intuitive to me so if there's something obvious that I failed to notice, don't be upset.

Thank you :)

Answer 1

I have included a graph for you for easy visualization:

Following the example you have given, $(x+3)(x+2)=0$ translates to you subbing $y=0$ in your equation. That means that you are looking for the $x$-coordinates that lie on the $x$-axis.

Therefore, you will obtain $x=-3$ and $x=-2$ as the solution. Remember, when you break down the equation $(x+3)(x+2)=0$, it gives

$x+3=0$ and $x+2=0$

When you bring over the value to the other side of the equation, remember to change the sign accordingly.

Answer 2

In a field, $ab=0$ implies either $a=0$, or $b=0$. Now, for $a=x+3$ and $b=x+2$, this says, that $(x+3)(x+2)=0$ implies $x+3=0$ or $x+2=0$. The quadratic equation is $x^2+5x+6=0$, and we see that indeed $x=-3$ and $x=-2$ are the two solutions.

Answer 3

In multiplication there is only on case that gives 0 as a result and it is when at least one of the sides of the multiplication is 0 . Since we dealing with variables here, we don't know which side is egal 0 , so we consider all the cases.

Answer 4

If you multiply a whole bunch or things together and the product is $0,$ then one of the factors that you multiplied must equal $0.$

$ab = 0$ if and only if $a=0$ or $b=0$

if $(x+3)(x+2)=0$ then $x+3 = 0$ or $x+2 = 0$