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Quadratic equations form the backbone of many algebraic problems. A quadratic equation is generally in the form of \[ax^2 + bx + c = 0\], where _a_, _b_, and _c_ are constants
with _a_ ≠ 0.

For example, the equation \[x^2 = 11\]
is a simple quadratic equation where _a_ = 1, _b_ = 0, and _c_ = -11. Solving quadratic equations often involves finding solutions for _x_
that satisfy the equation. These solutions are known as the roots of the equation.

There are several methods for solving quadratic equations, including factoring, using the quadratic formula, or completing the square.
Factors can be utilized when the quadratic expression can be simplified into two binomials.
The quadratic formula, \[x = \frac {-b ± \sqrt {b^2 - 4ac}}{2a} \],
can always be used to find real or complex solutions, and is especially useful when factoring is difficult. Finally, completing the square offers another route to simplify and solve quadratic equations.

In our specific exercise, we need to find values of _x_
such that \[x^2 = 11\]. This means we can directly find the square roots, giving us potential solutions.

Square roots are mathematical operations that return a value which, when multiplied by itself, results in the original number. For instance, the square root of \[11\]
is a number _x_
such that \[x^2 = 11\].

There are both positive and negative square roots because both values, when squared, will produce a positive number. So, \[\sqrt {11} and -\sqrt {11}\]
both satisfy the equation \[x^2 = 11\].

Some square roots result in whole numbers, such as \[\sqrt {9} = 3\],
but others can be irrational numbers that continue infinitely without repeating, like \[\sqrt {11}\].
When dealing with quadratic equations, recognizing how to take square roots can simplify the process of solving for _x_.

To find all solutions of the given function \[f(x) = x^2 = 11\],
we take the square root of 11 to determine the positive and negative values that satisfy the equation.

Solving equations involves finding the values of variables that make the equation true. In our case, we have the quadratic equation \[x^2 = 11\]
, and we need to determine the values of _x_.
By taking the square root of both sides, we determine: \[\sqrt {x^2} = \sqrt {11}\]
which simplifies to \[x = ±\sqrt {11}\].

This gives us two solutions: \[x = \sqrt {11}\]
and \[x = -\sqrt {11}\].

Sometimes, equations may have complex solutions, especially when taking the square root of negative numbers. For example, \[x^2 = -11\]
would give us \[x = ±\sqrt {-11}\]. \[_To simplify this_, \sqrt {-11}\]
is represented as \[\sqrt {11} \times \sqrt {-1}\],
which we know as \[i \sqrt {11}\]
(using \[i\] as the imaginary unit where \[i^2 = -1\])).

Understanding these fundamental steps will help you solve a variety of quadratic equations and handle both real and imaginary solutions with confidence.