Question

How do you factor \[{a^2} + ab - 2{b^2} + 2a - 2b\]?

Answer 1

Hint: Break the value of square of b into two parts. We write the given equation in such a way that we can pair the terms \[{a^2} - {b^2}\], now we take common values from terms and combine the factors. Open the term using the identity \[{a^2} - {b^2} = (a - b)(a + b)\]
* Factor of an equation means that the equation is exactly divisible by that factor. If there is one factor of an equation, then there can be other factors as well which when multiplied to each other give us the original equation.

Complete step-by-step answer:
We have to find factors of \[{a^2} + ab - 2{b^2} + 2a - 2b\]
We can take negative sign outside square of b
\[ \Rightarrow {a^2} + ab - \left( {2{b^2}} \right) + 2a - 2b\]
Now we can break the square of b
\[ \Rightarrow {a^2} + ab - \left( {{b^2} + {b^2}} \right) + 2a - 2b\]
Open the bracket and write the terms
\[ \Rightarrow {a^2} + ab - {b^2} - {b^2} + 2a - 2b\]
So, we can write the given equation as \[{a^2} + ab - {b^2} - {b^2} + 2a - 2b\]
Shuffle the terms and write the equation bringing \[{a^2} - {b^2}\] together
\[ \Rightarrow \left( {{a^2} - {b^2}} \right) + ab - {b^2} + 2a - 2b\]
Now we open the first bracket using the identity \[{a^2} - {b^2} = (a - b)(a + b)\]
\[ \Rightarrow \left( {a - b} \right)\left( {a + b} \right) + ab - {b^2} + 2a - 2b\]
Take b common from second and third term and 2 common from last two terms.
\[ \Rightarrow \left( {a - b} \right)\left( {a + b} \right) + b\left( {a - b} \right) + 2\left( {a - b} \right)\]
Now since all the terms have \[\left( {a - b} \right)\] common, we take this factor common
\[ \Rightarrow \left( {a - b} \right)\left[ {a + b + b + 2} \right]\]
Add the terms in bracket
\[ \Rightarrow \left( {a - b} \right)\left( {a + 2b + 2} \right)\]

\[\therefore \]The factors of \[{a^2} + ab - 2{b^2} + 2a - 2b\] are \[\left( {a - b} \right);\left( {a + 2b + 2} \right)\].

Note:
Many students make the mistake of leaving the factors after taking common factors like as it is, which is wrong. You have to write the simplest form of the other factor which is obtained from collecting the values after taking the common factor. Keep in mind factors divide the given equation completely, i.e. when divided by the product of factors the equation gives answer 0. Also, many students try to take common terms from the direct given equation, keep in mind we can shuffle the terms as per our requirement so as to form a pattern.