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 the following:
$Y=x^2+c$
For example:
$X^2+4=0$
In equations of this type, we won't need to use the quadratic formula or other long methods. Instead we can use a much more efficient and quicker method- by calculating the square root!

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

Move the terms and isolate $x^2$ so it will be alone on one side (even without a coefficient).
Perform the following calculation:
$x^2=16$

Let's take the square root of both sides and remember to insert $±$ due to the fact that when we take the square root of a free number there are $2$ opposite solutions (one negative and one positive)
Let's solve the following:
$x=±4$

Let's write the answers that we obtained in an organized way.
Proceed as follows:
$X_1=4$
$X_2=-4$

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 the following:
$3x^2=9$
$x^2=3$

Now, after we completely isolated $x^2$, we can take the square root of both sides. Let's not forget to place the $\pm$ sign
Continue to solve:
$x=±\sqrt3$
Let's proceed to 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:
To begin with, we'll move terms in order to isolate $X^2$
We'll perform the following:
$x^2=-4$
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. The square root of a negative number cannot be found.

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

Solution:
At first glance, this exercise might look slightly intimidating given that its numbers are relatively large. However, we can solve exercises like these easily using the method we've learned so far.
To begin with, we completely isolate $x^2$.
We perform the following:
$12X^2=240$
Divide both sides by $12$ as follows:
$x^2=20$
Now we'll take the square root and not forgetting to insert the $\pm$ sign
We'll perform the following:
$X=±\sqrt{20}$
Write both answers in a concise way:
$X_1=-\sqrt{20}$

$X_2=\sqrt{20}$

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

Solution:
To begin with let's move terms to obtain the following:
$3x^2=-150$
Let's proceed to divide both sides by $3$ in order to completely isolate the $X^2$.
We obtain the following:
$X^2=-50$
At this point, we can observe that the equation has no solution due to the fact that it's impossible for any number squared (whether positive or negative) to give us a negative number like $-50$.
Nevertheless, let's continue to prove that there's no solution and try to take the square root.
We obtain the following:
$x=±\sqrt{(-50)}$
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 obtain any solution when we need to find the square root of a negative number.

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

Solution:
To begin with, we'll move terms to obtain the following:
$3x^2=4$
Now we'll divide both sides by $3$ :
$x^2=\frac43$
Note, don't be alarmed by getting a fraction, we'll continue as usual. The next step is to take the square root. We obtain the following:
$x=±\sqrt{\frac43}$
We'll the proceed to 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.