Question

How do you solve \[{x^3}\, - \,{x^2}\, - \,\,12x\, = \,0\] ?

Answer 1

Hint: Try to solve it using Quadratic factorization using splitting of middle term method. In this method the middle term is split into two factors. Initially take the \[x\] common because it will become easier to solve.

Complete step by step answer:
Here for the given equation take \[x\] common. This will make the equation simpler and easier to solve.
\[x({x^2}\, - \,x\, - \,12)\, = \,0\]
Now solve the equation part by Quadratic factorisation using splitting of the middle term.
\[ \Rightarrow x({x^2}\, - 4x\, + \,3x\, - \,12)\, = \,0\]
\[ \Rightarrow x(x\, - \,4)\,\,(x\, + \,3) = \,0\]
\[ \Rightarrow (x\, - \,4)\,(x\, + \,3)\, = \,0\]
Now, use the null law method. In this method we need to set the first group and set it equal to 0 and then solve. For solving quadratic equations this law can be used. In this law a mathematical expression is used to be converted into an equation by making the equation equal to 0, and it becomes easy to get the answer after that.
\[x\, - \,4\, = \,0\]
\[ \Rightarrow x\, = \,4\]
Set the next group equal to 0 and solve,
\[x\, + \,3\, = \,0\]
\[ \therefore x\, = \, - 3\]
The final answer to the question is that \[x\, = \,0,4, - 3\]

Additional information:
Factorisation is a method in which a mathematical expression or a polynomial is divided into different terms so that it becomes easier to solve the expression. In the factorisation method, we reduce any algebraic or quadratic equation into its simpler form.

Note: Middle term splitting method is easy to find factors, but there is another method that is taking the highest common factor directly and obtaining the factors. This method is working for some questions only, mostly for 3 variable questions. This method makes the question solving very quick and is easy to use.