Step 1: Foil (multiply) the right hand side of the equation
#y= (x+5)(x+3)#
#rArr y= x^2 + 5x + 3x + 15#
#=> color(red)(y= x^2 + 8x + 15)#
Step 2: We can write the vertex form by several methods
Reminder: vertex form is #color(blue)(y= a(x-h)^2 + k)#
# =># Method 1: By completing square
#=> color(red)(y= x^2 + 8x + 15)# #=># re-write
We make a perfect trinomial in the form of
#=> a^2 -2ab +b^2 = (a-b)^2#
#=> a^2 +2ab+b^2 = (a+b)^2#
#y = (x^2 + 8x+color(green)16) color(green)(-16)+15#
#16= [1/2 (8)]^2#
#y= (x+4)^2 -1# Vertex form completed
# =># Method 2: Using formula
#h= x_(vertex)= -b/(2a)#
#k= y_(vertex)= y(-b/(ab))#
From this#=> color(red)(y= x^2 + 8x + 15)#
We have #a= 1# ; #b= 8# , #c= 15#
#h= x_(vertex)= -8/(2*2) = color(red)-4#
#k= y_(vertex)= y(-4) = (-4)^2+8(-4)+15#
#y(-4) = 16-32+15 =color(red) (-1) #
vertex form is #color(blue)(y= 1(x-(-4))^2 + (-1))#
simplify #color(red)(y= 1(x+4))^color(red)2-1#
