Question

Solve by factorization: \[{x^2} + x - 20 = 0\]

Answer 1

Hint:
Here, we will use the concept of factorization. Factorization is the process in which a number is written in the form 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 in the equation.

Complete step by step solution:
Given equation is \[{x^2} + x - 20 = 0\].
First, we will split the middle term into two parts such that its multiplication will be equal to the product of the first term and the third term of the equation. Therefore, we get
\[ \Rightarrow {x^2} + 5x - 4x - 20 = 0\]
Now we will be taking \[x\] common from the first two terms and taking 4 common from the last two terms. Therefore the equation becomes
\[ \Rightarrow x\left( {x + 5} \right) - 4\left( {x + 5} \right) = 0\]
Now we will take \[\left( {x + 5} \right)\] common from the equation. Therefore, we get
\[ \Rightarrow \left( {x + 5} \right)\left( {x - 4} \right) = 0\]

Hence, \[\left( {x + 5} \right),\left( {x - 4} \right)\] are the factors of the given equation \[{x^2} + x - 20 = 0\].

Note:
The given equation is a quadratic equation and by the definition of the quadratic equation the highest degree of the variable is 2. Also, a quadratic equation has two solutions or roots of the equation. Here we should split the middle term very carefully according to the basic condition. The 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 multiplication 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.