Question

Factorize: \[{x^2} + 10x + 24\].

Answer 1

Hint: Here, we will use the concept of the factorization. Factorization is the process in which a number is written in the forms of its small factors which on multiplication give the original number. First, we will split the middle term of the equation and then we will form the factors by taking the common terms from the expression.

Complete step by step solution:
The given expression is \[{x^2} + 10x + 24\].
The given expression is a quadratic expression, so we will use a middle term splitting method.
First, we will split the middle term into two parts such that its product will be equal to the product of the first term and the third term of the expression. Therefore, we get
\[ \Rightarrow {x^2} + 10x + 24 = {x^2} + 6x + 4x + 24\]
Now taking \[x\] common from the first two terms and taking 4 common from the last two terms, the equation becomes
\[ \Rightarrow {x^2} + 10x + 24 = x\left( {x + 6} \right) + 4\left( {x + 6} \right)\]
Now we will take \[\left( {x + 6} \right)\] common from the equation. Therefore, we get
\[ \Rightarrow {x^2} + 10x + 24 = \left( {x + 6} \right)\left( {x + 4} \right)\]

Hence after factorization of the given equation we get the factors as \[\left( {x + 6} \right)\] and \[\left( {x + 4} \right)\].

Note:
Here we will split the middle term according to the basic condition. The basic condition is that the middle term i.e. term with the single power of the variable should be divided in such a way that its product must be equal to the product of the first and the last term of the equation. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations. Generally in these types of questions algebraic identities can be used to solve and make the factors.