We are given the quadratic equation:
#color(blue)(v^2-95v-96) = 0# #.. color(red)(Equation.1)#
Our quadratic expression is
#color(blue)(v^2-95v-96)# #.. color(red)(Expression.1)#
To factor this quadratic expression, we will follow the procedure given below:
#color(green)(Step.1)#
We must split the coefficient of middle term into two numbers , such that when we add them we get the middle term, and when we multiply them we must get the product of the coefficient of the #x^2 term# and the constant,
Note that the product of the coefficient of the #x^2 term# and the constant is #(-96)#,
#color(green)(Step.2)#
The two numbers are: #color(blue)(-96 and +1)#
When we add ( - 96) and ( +1 ) we get #(- 95)# and when we multiply the two values ( - 96) and ( +1 ) we get ( - 96 )
Now, we write our #.. color(red)(Expression.1)# as follows:
#color(blue)(v^2-96v + 1v-96)# #.. color(red)(Expression.2)#
#color(green)(Step.3)#
In this step, we break our #.. color(red)(Expression.2)# into groups:
#color(blue)(rArr (v^2 - 96v) + (1v - 96))#
Factor out #color(green)(v)# from #color(blue)((v^2 - 96v)# to obtain #color(blue)(rArr v*(v - 96) )#
Factor out #color(green)(1)# from #color(blue)((1v - 96)# to obtain #color(blue)(rArr 1*(v - 96) )#
#color(green)(Step.4)#
Using #color(green)(Step.3)# we can factor out the common term #color(blue)((v+1)# and write the factors of our quadratic expression:
#color(blue)(rArr (v + 1) * (v - 96)#
#color(green)(Step.5)#
Now, we are in a position to consider
#color(blue)(v^2-95v-96 = 0)# #.. color(red)(Equation.1)#
We can use the factors and write it as:
#color(blue)(rArr (v + 1) * (v - 96) = 0#
#color(blue)(rArr (v + 1) = 0 or (v - 96) = 0#
#color(blue)(rArr (v = - 1) or (v = 96)#
So our final solution set is given by
#color(blue)(rArr v = - 1, v = 96#
I hope this helps.
