Question

How to factorize the equation $3{x^2} - 10x + 3$?

Answer 1

Hint: For solving this equation we should know about the factorization. The process of writing a number as a product of several factors. When factors are multiplied together then they form the original number as in the beginning.

Complete step-by-step solution:
The most common method of factorization is to completely factor the number into its positive prime factors. The number which has only positive factor1 and the number itself, is termed as the prime number.

The factors of any given equation may be an integer, an algebraic expression or a variable or itself.
 For example, 9 and 5 are the factors of 8. There are six methods to solve factorization. These are as follows:- Greatest Common Factor, Grouping Method, Sum or difference in two cubes, Difference in two squares method, General trinomials and Trinomial method.

Now we will solve the following equation by the use of factorization method.

$\Rightarrow 3{x^2} - 10x + 3$

\[\Rightarrow 3{x^2} - 9x - x + 3\]

$\Rightarrow 3x \left( {x - 3} \right) - 1\left( {x - 3} \right)$

Now by taking out the common factor we get

$\Rightarrow \left( {x - 3} \right)\left( {3x - 1} \right)$

Now when we will put this value equal to zero we get:-

For the value $\left( {x - 3} \right)$

$\Rightarrow x - 3 = 0\,\,\, or\,\,\,\, x = 3$

Similarly for the value $\left( {3x - 1} \right)$

$\Rightarrow 3x - 1 = 0\,\,\,\,\, or \,\,\,\,\, x = \dfrac{1}{3}$

Thus, the factors of $3{x^2} - 10x + 3$ are $3\,\,\,and\,\,\,\dfrac{1}{3}$

Note: Factorization is an important process, which helps us understand more about the equations. Through factorization, usually we rewrite the polynomials in its simpler form, and when we use factorization’s principles in any equation, we find a lot of useful information about it. The ancient Greek Mathematicians were the first one who considered the factorization in the case of integers.