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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.