How do you solve ${{x}^{2}}-361=0$ ?

Hint: We are given ${{x}^{2}}-361=0$ to solve this, we learn about the type of equation we are given then learn the number of solutions of the equation. We will learn how to factor the quadratic equation, we will use the middle term split to factor the term and we will simplify by taking common terms out. We also use zero product rules to get our answer. To be sure about your answer we can also check by putting the acquired value of the solution in the given equation and check whether they are the same or not.

Complete step-by-step solution:
We are asked to solve the given problem ${{x}^{2}}-361=0$.
First we observe that it has a maximum power of ‘2’ so it is a quadratic equation.
Now we should know that a quadratic equation has 2 solutions or we say an equation of power ‘n’ will have an ‘n’ solution.
Now as it is a quadratic equations we will change it into standard form $a{{x}^{2}}+bx+c=0$
We have ${{x}^{2}}-361=0$.
We can write it in standard form as –
By simplifying, we get –
${{x}^{2}}-361=0$
Now we have to solve the equation ${{x}^{2}}-361=0$.
To solve this equation we first take the greatest common factor possibly available to the terms.
As we can see that in ${{x}^{2}}-361=0$
1 and 361 have nothing in common. So the equation remains the same.
${{x}^{2}}-361=0$
We know that 361 is the square of 19 i.e. $361={{19}^{2}}$ . So, we can write the above equation as –
$\Rightarrow {{x}^{2}}-{{19}^{2}}=0$ .
 We also know that ${{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)$ .
So we apply this on ${{x}^{2}}-{{19}^{2}}=0$ and we will get –
$\Rightarrow \left( x-19 \right)\left( x+19 \right)=0$
Using zero product rule, which says that if product of two terms is zero then either one of them is zero.
So,
Either $x-19=0$ or $x+19=0$ .
So we get –
$x=19$ and $x=-19$
Hence the solution are $x=19,-19$.

Note:
We can also solve this using a method which is called by factoring, we factor using middle term split.
For quadratic equations $a{{x}^{2}}+bx+c=0$ we will find such a pair of terms whose products are the same. As $a\times c$ and whose sum or difference will be equal to the b.
We can cross check our solution.
We put our value in the given Equation and if it satisfies the equation then it is the correct value.
We put $x=19$ in ${{x}^{2}}-361=0$.
We get ${{19}^{2}}-361=0$ .
$361-361=0$
$0=0$ .
So, the equation is satisfied.
Hence, it is the correct solution.
Now secondly,
We put $x=-19$ in ${{x}^{2}}-361=0$.
We get ${{\left( -19 \right)}^{2}}-361=0$ .
$361-361=0$ .
$0=0$
So, the equation is satisfied.
Hence it is correct solution
So, our solution is correct.