#" "#
We are given the quadratic function in Standard Form:
#color(red)(y=f(x)=-3x^2+17x+2#
#color(blue)(y=f(x)=ax^2+bx+c#
What is expected?
We must convert to Vertex Form:
#color(blue)(y=f(x)=a(x-h)^2+k#
We have,
#y=f(x)=-3x^2+17x+2#
#color(green)("Step 1"#
Use Completing the Square Method to convert to Vertex Form:
#rArr (-3x^2+17x)+2#
#rArr -3[x^2-(17/3)x]+2#
#color(green)("Step 2"#
#rArr -3[x^2-(17/3)x+ square]+2#
In the #square# above, add #[(1/2)(17/3)]^2#
#rArr -3[x^2-(17/3)x+ [(1/2)(17/3)]^2]+2#
#rArr -3[x^2-(17/3)x+ (17/6)^2]+2#
#color(green)("Step 3"#
#rArr -3[x^2-(17/3)x+ (17/6)^2]+2- square#
Since we added #(17/6)^2# in the previous step, we must also subtract the same value.
#rArr -3[x^2-(17/3)x+ (17/6)^2]+2- (17/6)^2#
#color(green)("Step 4"#
On simplification, we get
#rArr -3[x^2-(17/3)x+ (17/6)^2]+(939/36)#
#y=f(x)= -3[x-17/6]^2+(939/36)#
Now, we have the required vertex form.
Hope this helps.
