Question

How do you factor the equation $ 3{x^2} + 13x - 30? $

Answer 1

Hint: To factorize the given equation, first of all use the sum product method to split the one degree term into two terms, such that the product of terms will be equal to the product of two degree terms and the constant. Then you will get four terms, make their pairs and from each pair take out the common term. Now, rewrite the equation in factored form.

Complete step by step solution:
In order to factorize the given expression $ 3{x^2} + 13x - 30 $ we will use sum product method to split the middle term into two terms as follows
Since the product of two split terms should be equals to the product of two degree term and the constant, so it will be easy to find split terms if we find the factors of $ 3 \times ( - 30) = - 90 $
 $ - 90 = - 1 \times 2 \times 3 \times 3 \times 5 $
Here if we take $ 2 \times 3 \times 3 = 18\;{\text{and}}\; - 1 \times 5 = - 5 $ then we will get the required terms,
Hence $ 18 + ( - 5) = 13 $
So rewriting the expression as
 $
   = 3{x^2} + 13x - 30 \\
   = 3{x^2} + 18x - 5x - 30 \;
  $
Now we will make pair of first two terms and last two terms, and then take out the common factor from the pairs, we will get
 $
   = \left( {3{x^2} + 18x} \right) - \left( {5x + 30} \right) \\
   = 3x\left( {x + 6} \right) - 5\left( {x + 6} \right) \;
  $
Now taking $ \left( {x + 6} \right) $ common in the expression, we will get
  $ = \left( {x + 6} \right)\left( {3x - 5} \right) $
Therefore, $ \left( {x + 6} \right)\left( {3x - 5} \right) $ is the required factored form of the given expression.
So, the correct answer is “ $ \left( {x + 6} \right)\left( {3x - 5} \right) $ ”.

Note: There is an alternative method to do this factorization, first hit and trial, in this method you have to find one of the factors by putting some random values in the expression to get zero then after divide the expression with that factor to get other factors and finally write the expression as the product of all the factors.