Question

How do you factor $ 7a{b^2} - 77ab + 168a $ ?

Answer 1

Hint: First of all find the common multiples from all the terms and then use the concept of splitting the middle terms and first making the pair of two terms and then finding the common factors from the paired terms and finally the factors for the given expression.

Complete step-by-step answer:
Take the given expression: $ 7a{b^2} - 77ab + 168a = 0 $
Take the common factor from all the three terms.
 $ \Rightarrow 7a({b^2} - 11b + 24) = 0 $
Term multiplicative on one side if moved to the opposite side, then it goes to the denominator.
 $ \Rightarrow ({b^2} - 11b + 24) = \dfrac{0}{{7a}} $
Zero upon any number is zero.
 $ \Rightarrow {b^2} - 11b + 24 = 0 $
Here now split the middle term and find the factors.
To split the middle term, we will split in such a way that the product of the two split terms will be equal to the product of first and the last terms. Also, with the same sign convention.
When you have to simplify one negative term and one positive term then for the resultant value you have to perform subtraction and sign of a bigger digit. In multiplication if there is one negative and one positive term then the resultant product will be with negative sign.
So,
 $
   - 11 = - 3 - 8 \\
   + 24 = ( - 3)( - 8) \;
  $
 Take the simplified expression,
 $ \Rightarrow {b^2} - 11b + 24 = 0 $
The above expression can be re-written by splitting the middle term,
 $ \Rightarrow {b^2} - 8b - 3b + 24 = 0 $
Make pair of first two and last two terms,
 $ \Rightarrow \underline {{b^2} - 8b} - \underline {3b + 24} = 0 $
Take out common factors from the above two paired terms.
 $ b(b - 8) - 3(b - 8) = 0 $
Take out common from the above equation, Take common multiple common from the first bracket from both the terms.
 $ (b - 8)(b - 3) = 0 $
 $
   \Rightarrow b - 8 = 0 \\
   \Rightarrow b = 8 \;
  $
Or
 $
   \Rightarrow b - 3 = 0 \\
   \Rightarrow b = 3 \;
  $
Hence, the solutions are $ b = 3,8 $
So, the correct answer is “$ b = 3,8 $”.

Note: Here we were able to split the middle term and find the factors but in case it is not possible then we can find factors by using the formula\[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\] and considering the general form of the quadratic equation $ a{x^2} + bx + c = 0 $ . Be careful about the sign convention and simplification of the terms in the equation.