What are the different methods for solving quadratic equations?

Hint: There are different methods to solve a quadratic equation. There is a method in which we split the middle term. We can also use a formula which is called a quadratic formula to solve the quadratic equation. There is one more method which we called as completing the square method.

Complete step-by-step solution:
A quadratic equation makes a ∪-shaped curve (parabola) if we represent it graphically. There are always two solutions to a quadratic equation with either real roots or imaginary roots.
There are some methods to solve the quadratic equation. They are,
1. Factorization method
2. Completing square method
3. Formula method
Solving Quadratic Equation by Factorization Method
If we can factorize \[a{x^2} + bx + c\,,\,\,a \ne 0\], into a product of two linear factors, then the roots of the quadratic equation \[a{x^2} + bx + c = 0\] can be found by equating each factor to zero.
For example,
$ \Rightarrow \;{x^2} + 5x + 6 = 0$
Now, splitting the middle term
$ \Rightarrow {x^2} + (2 + 3)x + 6 = 0$
\[ \Rightarrow {x^2} + 2x + 3x + 6 = 0\]
Taking common,
\[ \Rightarrow x(x + 2) + 3(x + 2) = 0\]
\[ \Rightarrow (x + 2)(x + 3) = 0\]
At last, we will use the zero product rule to find the zeros. Zero product property says that when \[p \times q = 0\] then either \[p = 0\,\,or\,\,q = 0\]
Therefore, \[(x + 2) = 0,\,\,\,or\,\,(x + 3) = 0.\]
Hence, the solutions or roots of the quadratic equation \[{x^2} + 5x + 6 = 0\,\,\,\;are\,\;x = -2\,,\,\,x = -3.\]
Solving Quadratic Equation by Completing the Square Method
A quadratic equation can be solved by the method of completing the square. In this method we make the whole square by adding and subtracting numbers.
Solving \[{x^2}-{\text{ }}6x{\text{ }}-{\text{ }}3 = {\text{ }}0\] by using completing square method formula –
\[ \Rightarrow {x^2}-6x-3 = 0\]
\[ \Rightarrow {x^2}-6x = 3\]
Now, adding ${\left( { - 3} \right)^2}$both sides
\[ \Rightarrow {x^2} - 6x + {\left( { - 3} \right)^2} = 3 + 9\]
Using formula ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
\[ \Rightarrow {\left( {x - 3} \right)^2} = 12\]
\[ \Rightarrow x - 3{\text{ }} = {\text{ }} \pm {\text{ }}\surd 12\]
\[ \Rightarrow x = 3 \pm 2\surd 3\]
Solving Quadratic Equation by Formula Method
By learning the quadratic equation formula, you can solve any quadratic equation quickly. If the quadratic equation looks like this, then below is the formula you need to apply.
$ \Rightarrow \,x\, = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Let’s take an example, \[27{x^2} - {\text{ }}12{\text{ }} = {\text{ }}0\]
Here, \[a{\text{ }} = {\text{ }}27,{\text{ }}b{\text{ }} = {\text{ }}0{\text{ }}and{\text{ }}c{\text{ }} = {\text{ }} - 12.\;\]
Now, by putting the values in the quadratic equation formula, you get:
$ \Rightarrow \,x\, = \,\dfrac{{ - 0 \pm \sqrt {{0^2} - 4(27)( - 12)} }}{{2(27)}}$
On simplification, we get
$ \Rightarrow x\, = \pm \,\sqrt {\dfrac{4}{9}} $
Finally, $x\, = \, \pm \,\dfrac{2}{3}$.

Note: These are some few methods to solve a quadratic equation. There is one method called the graphical method of solving quadratic equations. In this method wherever the curve of our equation cuts the x-axis determines the roots of our equation.