Question

Factorise:
 $ 3{x^2} + 2x - 5 $

Answer 1

Hint: Factorisation means that we have to find the factors constituting the given equation. For that, we can write the middle term as either the sum or difference between the factors of the product of the first and the last term. Then we can group together the like terms, take out the common terms and then obtain the required factors.

Complete step-by-step answer:
The given equation for the factorisation is $ 3{x^2} + 2x - 5 $
As the maximum degree of this equation is 2, we can call this as a quadratic equation.
Factorisation is basically simplification of this expression and the given expression can be simplified as:
We need to find the product of the constant and the coefficient of $ {x^2} $ and then use the sum or product of its factors to make it equal to the middle term i.e. the coefficient of x.
Product = $ 3 \times ( - 5) = - 15 $
The factors of 15 are:
 $ 15 = 1 \times 3 \times 5 $
1, 3 and 5, we have to get 2 by either addition or subtraction of either two but with numbers that are present in the equation. Such numbers are 5 and 3.
The product is negative which means either of the numbers will be negative, but we need 21 positive so, the greater value can be positive and the smaller value will be negative.
 $ \Rightarrow 5x - 3x = 2x $
Substituting in the given equation, we get:
 $ 3{x^2} + 5x - 3x - 5 $
Getting the like terms together and taking out common values:
 $
  3{x^2} - 3x + 5x - 5 \\
  \Rightarrow 3x(x - 1) + 5(x - 1) \\
  \Rightarrow (3x + 5)(x - 1) ;
  $
Therefore the factors of $ 3{x^2} + 2x - 5 $ are (3x + 5) and (x – 1)

Note: We chose the factors that are already present in the equation so as to obtain the like terms and by which the common values can be easily taken.
We have to be careful about the signs of both the product and sum/division of the factors.
We can also find the value of the variable using these factors but when we are asked to factorise the equation, we just write the factors obtained.
For checking if we have found the correct factors or not, we can multiply both, if the original equation is obtained, the factors are right.