Solve the following quadratic equation: \[{x^2} + 2\sqrt 2 x - 6 = 0\]

Hint: Here we are asked to solve the given equation for the value of the unknown variable $x$ . We will use the technique of taking terms common from the given equation and then solve, then write the equation into simpler expressions and forms. Then after writing it into forms of factors in multiplied form, we will equate it to the right-hand side of the equation to find the values of the unknown variable. Since the given equation is of degree two the unknown variable will have two roots.

Complete answer:
We are given with the equation \[{x^2} + 2\sqrt 2 x - 6 = 0\]
Now we will use the middle term factorization to solve the quadratic equation,
\[{x^2} + 2\sqrt 2 x - 6 = 0\]
Now let us break the middle term that is $2\sqrt 2 x$
\[ \Rightarrow {x^2} + 3\sqrt 2 x - \sqrt 2 x - 6 = 0\]
Now we can see that the variable $x$ is common in the first two terms and the term $ - \sqrt 2 $ is common in the last two terms so let us take them commonly out from that.
\[ \Rightarrow x(x + 3\sqrt 2 ) - \sqrt 2 (x + 3\sqrt 2 ) = 0\]
As we can see that the term $x + 3\sqrt 2 $ is now common in both the terms so let us take it commonly out from that.
\[ \Rightarrow (x - \sqrt 2 )(x + 3\sqrt 2 ) = 0\]
From the above we get
$x - \sqrt 2 = 0,x + 3\sqrt 2 = 0$
On simplifying the above, we get
\[ \Rightarrow x = \sqrt 2 , - 3\sqrt 2 \]
Therefore, the solutions to the quadratic equations are \[x = \sqrt 2 , - 3\sqrt 2 \]

Note:
The degree of any equation can be easily found by finding the powers of the unknown variables. The highest power of the unknown variable gives the degree of that equation. Here we can see that the highest power of the unknown variable is two thus, the degree of the given equation is two.