Question

If 2x² + 7x + 10 = p(x+q)² +r for all values of x, find the values of p and q

Answer 1

If the coordinates of the vertex of a parabola are known
`V(p,k)`, the equation for that parabola can be written by the following formula;7

• 
  `y=a.(x-p)^2+k`

Now let's find the vertex of the given equation.

• 
  `V(p)=(-b)/(2a)=(-7)/(4) =p`
• 
  `V(k)=f(p)=2((-7)/(4) )^2+7((-7)/(4) )+10=(31)/(8) =k`

We've found our vertex.

• 
  `V(p,k)=((-7)/(4) ,(31)/(8) )`

Let's substitute the unknowns in the formula I've given.

• 
  `p(x+q)^2+r=p(x+(7)/(4) )^2+(31)/(8)`

Let's expand the expression and find the coefficient p.

• 
  `px^2+(7px)/(2)+(49p)/(16)+(31)/(8) =2x^2+7x+10`
• 
  `p=2`

Finally;

`q=(7)/(4)`

`p=2`