Solve the quadratic equation ${{x}^{2}}+5x+6=0$

Hint: We will solve the given quadratic equation by using the splitting the middle term method. We will split the middle term of the equation $a{{x}^{2}}+bx+c=0$ such that the product of two numbers is equal to $a\times c$ and the sum of two numbers is equal to $b$.

Complete step by step answer:
We have been given an equation ${{x}^{2}}+5x+6=0$.
We have to solve the given quadratic equation.
Now, we will use the split middle term method to solve the given equation. We have to find two numbers such that the product of two numbers is equal to $a\times c=1\times 6=6$ and their sum is equal to $b=5$.
So we will use two numbers as 3 and 2.
Now, splitting the middle term we will get
$\Rightarrow {{x}^{2}}+\left( 3x+2x \right)+6=0$
Now, simplifying the above obtained equation we will get
$\Rightarrow {{x}^{2}}+3x+2x+6=0$
Now, taking the common terms out we will get
$\Rightarrow x\left( x+3 \right)+2\left( x+3 \right)=0$
Now, again taking common factors out we will get
$\Rightarrow \left( x+3 \right)\left( x+2 \right)$
Now, equating each factor to zero we will get
$\Rightarrow \left( x+3 \right)=0$ and $\left( x+2 \right)=0$
Now, simplifying the above obtained equations we will get
$\Rightarrow x=-3$ and $x=-2$
Hence on solving the given quadratic equation we get the values of x as $-3,-2$.

Note: Here in this question we use the split middle term method as it is a simple question. We can also use other methods like quadratic formula, completing the square method also to solve the quadratic equations. The value of x is the solution of the quadratic equation. Number of solutions depends on the degree of the equation.