Question

How do you solve $ 49{b^2} + 84b + 32 = 0 $ by completing the square?

Answer 1

Hint: First of all observe the given expression and which suggests that first term is the perfect square and then will find the last term taking the given first and the middle terms and then converting the whole equation in the form of complete square and then will find the resultant required value for “b”.

Complete step by step solution:
Take the given expression: $ 49{b^2} + 84b + 32 = 0 $
The above equation suggests that the first term is the whole square of $ 7 $ .
Also, the term such as $ 7 \times A = \dfrac{{84}}{2} $
Simplifying the above expression, performing that term multiplicative on one side if moved to the opposite side then it goes to the denominator.
\[A = \dfrac{{84}}{{7 \times 2}}\]
\[A = 6\]
So, the last term should be the square of
Therefore, the given expression can be re-written as, adding and subtracting the same number.
 $ 49{b^2} + 84b + 36 - 36 + 32 = 0 $
Rearranging the above equation.
 $ \underline {49{b^2} + 84b + 36} - \underline {36 + 32} = 0 $
The above equation can be written as the whole square
 $ {(7b + 6)^2} - 4 = 0 $
Move the constant term on the opposite side. When you move any term from one side to another then the sign of the term also changes. Negative terms becomes positive.
 $ {(7b + 6)^2} = 4 $
Take the square root on both the sides of the equation.
 $ \sqrt {{{(7b + 6)}^2}} = \sqrt 4 $
 $ \sqrt {{{(7b + 6)}^2}} = \sqrt {{2^2}} $
Square and square root cancel each other.
 $ \Rightarrow 7b + 6 = \pm 2 $
We get two values –
 $ \Rightarrow 7b + 6 = 2 $ or $ \Rightarrow 7b + 6 = - 2 $
Simplify the equation making the variable “b” as the subject-
 $ \Rightarrow 7b = 2 - 6 $ or $ \Rightarrow 7b = - 2 - 6 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow b = - \dfrac{4}{7} $ or $ \Rightarrow b = - \dfrac{8}{7} $
This is the required solution.
So, the correct answer is “ $ b = - \dfrac{4}{7} $ or $ b = - \dfrac{8}{7} $ ”.

Note: Always remember that the square root of any positive term can be positive or the negative and square of positive or the negative number gives resultant value always positive. Be good in squares and square root numbers and remember at least till twenty for the efficient and the accurate solution.