The vertex form of a quadratic equation is #a (x - h)^(2) + k#.
We have: #y = (6 x + 3)(x - 5)#
To express this equation in its vertex form, we must "complete the square".
First, let's expand the parentheses:
#Rightarrow y = 6 x^(2) - 30 x + 3 x - 15#
#Rightarrow y = 6 x^(2) - 27 x - 15#
Then, let's factor #6# out of the equation:
#Rightarrow y = 6 (x^(2) - frac(27)(6) x - frac(15)(6))#
#Rightarrow y = 6 (x^(2) - frac(9)(2) x - frac(5)(2))#
Now, let's add and subtract the square of half of the #x# term within the parentheses:
#Rightarrow y = 6 (x^(2) - frac(9)(2) x + (frac(9)(4))^(2) - frac(5)(2) - (frac(9)(4))^(2))#
#Rightarrow y = 6 ((x - frac(9)(4))^(2) - frac(5)(2) - frac(81)(16))#
#Rightarrow y = 6 ((x - frac(9)(4))^(2) - frac(121)(16))#
Finally, let's distribute #6# throughout the parentheses:
#therefore = 6 (x - frac(9)(4))^(2) - frac(363)(8)#
