Question

Write in the form p(x+q)²+r


2x²-5x+1

Answer 1

`\text{Let}\ f(x)=ax^2+bx+c,\ \text{then}\ f(x)=a(x-p)^2+k,\ \text{where}\\\\h=(-b)/(2a),\ k=f(h)\\\\-----------------\\\\\text{We have}\ f(x)=2x^2-5x+1\to a=2,\ b=-5,\ c=1.\\\\\text{Substitute:}\\\\h=(-(-5))/(2(2))=(5)/(4)\\\\k=f\left((5)/(4)\right)=2\left((5)/(4)\right)^2-5\left((5)/(4)\right)+1=2\left((25)/(16)\right)-(25)/(4)+1\\\\=(25)/(8)-(25\cdot2)/(4\cdot2)+(8)/(8)=(25-50+8)/(8)=-(17)/(8)\\\\\text{Therefore we have}`

`f(x)=2\left(x-(5)/(4)\right)-(17)/(8)=\boxed{2\left(x+\left(-(5)/(4)\right)\right)^2+\left(-(17)/(8)\right)}`

`q=-(5)/(4),\ k=-(17)/(8)`