Question

How do you factor \[{{a}^{3}}-2{{a}^{2}}-4a=-8\]?

Answer 1

Hint: For this question we are asked to factor the equation \[{{a}^{3}}-2{{a}^{2}}-4a=-8\]. So, firstly we will bring everything to one side of the equation and we will factor out a common factor in the equation after rearrangement of the terms in the equation. By this way we will solve the given question.

Complete step by step solution:
Firstly, we will bring all the terms in the equation to one side that is the left hand side of the equation. So, the equation will be reduced as follows.
\[\Rightarrow {{a}^{3}}-2{{a}^{2}}-4a+8=0\]
So, after bringing the terms to one side we will rearrange the terms in the equation so that we can easily take the common factor common in the terms of the equation.
In the above equation we can clearly observe that from terms 1,2 we can take \[{{a}^{2}}\] common in the same way from the terms 3,4 we can take -4 as common.
So, the equation will be reduced as follows.
\[\Rightarrow {{a}^{2}}\left( a-2 \right)-4\left( a-2 \right)=0\]
Here we can observe that the term \[\left( a-2 \right)\] is common in the equation. So, we take \[\left( a-2 \right)\] as a common factor and simplify the equation. So, the equation after doing the above process will be reduced as follows.
\[\Rightarrow {{a}^{2}}\left( a-2 \right)-4\left( a-2 \right)=0\]
\[\Rightarrow \left( {{a}^{2}}-4 \right)\left( a-2 \right)=0\]
Here, we further solve the equation using the basic formula in algebra which is \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]. So, after using the formula the equation will be further simplified as follows.
\[\Rightarrow \left( a+2 \right)\left( a-2 \right)\left( a-2 \right)=0\]
Therefore, the factor for the given question is \[ \left( a+2 \right)\left( a-2 \right)\left( a-2 \right)=0\].
Where \[a=\pm 2\] is the solution set.

Note:
Students must be careful in doing the calculations and students must have knowledge in basic algebra and its formulae like,
\[\Rightarrow {{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\].
Students must be careful in taking the terms common if we take \[-2{{a}^{2}},-4a\] common then our whole solution will be wrong.