Solution by extracting a root

Pay attention -
This solution method is suitable for quadratic equations where there is no $b$, and $Y=0$, meaning quadratic equations that look like this:
$Y=x^2+c$
For example:
$X^2+4=0$
In equations of this type, we won't need to use the quadratic formula or other long methods, and we can use an efficient and quick way - taking the square root!

Let's learn through an example.
Let's solve together the equation:
$x^2-16=0$

Let's move terms and isolate $x^2$ so it will be alone on one side (even without a coefficient).
Let's perform:
$x^2=16$

Let's take the square root of both sides and remember to put $±$ because when taking the square root of a free number there are $2$ opposite solutions (one negative and one positive)
Let's solve:
$x=±4$

Let's write the answers we got in an organized way.
We will perform:
$X_1=4$
$X_2=-4$

Excellent! Now we'll continue practicing solving equations with one variable $1$, increase the difficulty level and encounter different cases.

Exercise with coefficient greater than $1$ for $x^2$:
$3x^2-9=0$

Solution:
Let's start by completely isolating $x^2$. We'll perform:
$3x^2=9$
$x^2=3$

Now, after we completely isolated $x^2$, we can take the square root of both sides. Let's not forget to put the $\pm$ sign
Let's solve:
$x=±\sqrt3$
Let's write the answers in an organized way:
$X_1=\sqrt3$
$X_2= -\sqrt3$

Exercise requiring finding the square root of a negative number:
$X^2+4=0$

Solution:
First, we'll move terms to isolate $X^2$
We'll perform:
$x^2=-4$
Now we need to find the square root as usual. But!! We cannot find the square root of a negative number!
This means that there is no number that when squared will give us a negative result.
Therefore when we try to find the square root:
$X=\sqrt{(-4)}$
The answer will be - no solution!
Note! If you leave the answer like this
$X=\sqrt{(-4)}$
it would be a mistake. You must write - no solution. Square root of a negative number cannot be found.

Another exercise:
$12x^2-240=0$

Solution:
At first glance, this exercise might look a bit intimidating because its numbers are relatively large. However, we can solve exercises like these easily using the method we've learned so far.
First, we'll completely isolate $x^2$.
We'll perform:
$12X^2=240$
We'll divide both sides by $12$ and get:
$x^2=20$
Now we'll take the square root and not forget the $\pm$ sign
We'll perform:
$X=±\sqrt{20}$
We'll write both answers in an organized way:
$X_1=-\sqrt{20}$

$X_2=\sqrt{20}$

Another exercise:
$3x^2+150=0$

Solution:
First, let's move terms to get:
$3x^2=-150$
Now let's divide both sides by $3$ to completely isolate $X^2$.
We get:
$X^2=-50$
At this point, we can already see that the equation has no solution because it's impossible for any number squared (whether positive or negative) to give us a negative number like $-50$.
Nevertheless, let's continue to show there's no solution and try to take the square root.
We get:
$x=±\sqrt{(-50)}$
The answer will be - no solution since we cannot take the square root of a negative number.

Important note –
Sometimes students get confused and tend to think there is only $1$ solution to the equation, the solution of the $+$.
This is a serious mistake since we still need to find the square root of $50-$ and not of $50$, therefore we won't get any solution when we need to find the square root of a negative number.

Exercise with a fractional result:
$3X^2-4=0$

Solution:
First, we'll move terms to get:
$3x^2=4$
Now we'll divide both sides by $3$ to get:
$x^2=\frac43$
Note, don't be alarmed by getting a fraction, we'll continue as usual. The next step is to take the square root. We get:
$x=±\sqrt{\frac43}$
And we'll write the results in an organized way:
$X_1=\sqrt{\frac43}$
$X_2=-\sqrt{\frac43}$

Important note - when a square root can be easily simplified, like for the number $\sqrt{16}$ for example, you should not leave the answer like that and write $4$.
However, if you cannot get a whole number square root, it is usually fine to leave the answer with the square root.