Question

How do you factor and solve \[2{x^2} + x - 1 = 0\].

Answer 1

Hint:Here, we will use the concept of the factorization. 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. Then we will equate the terms to zero to get the value of \[x\].

Complete step by step solution:
Given equation is \[2{x^2} + x - 1 = 0\].
Factorization is the process in which a number is written in the form of its small factors which on multiplication give the original number.
Firstly 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
\[2{x^2} + 2x - x - 1 = 0\]
Now we will be taking \[x\] common from the first two terms and taking \[ - 1\] common from the last two terms. Therefore the equation becomes
\[ \Rightarrow 2x\left( {x + 1} \right) - 1\left( {x + 1} \right) = 0\]
Now we will take \[\left( {x + 1} \right)\] common from the equation we get
\[ \Rightarrow \left( {x + 1} \right)\left( {2x - 1} \right) = 0\]
Now as the above equation is equal to zero therefore, the factors will be equal to zero. Therefore, we get
\[ \Rightarrow \left( {x + 1} \right) = 0\] or \[\left( {2x - 1} \right) = 0\]
Now by solving this we will get the value of \[x\]. Therefore, we get
\[ \Rightarrow x = - 1\] or \[x = \dfrac{1}{2}\]

Hence by solving the equation \[2{x^2} + x - 1 = 0\] we will get the value of \[x\] as \[x = - 1\] and \[x = \dfrac{1}{2}\].

Note: Here we will note that we should split the middle term by using the basic condition that 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. We should know that the factors we have obtained on solving that we will get the originally given equation. Factors are the smallest part of the number or equation which on multiplication will give us the actual number of equations.