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Quadratics - max/min values

Hi everyone. Just looking at my revision checklist for the higher paper, and there's something listed that says 'completing the square/maximum values' ... I've got no idea how these could be connected. can anyone help please? :confused:

Reply 1

yusufu

When you have completed square form:

(x+a)² + b

the mininum point is at (-a,b).

When it's:

-(x+c)² + d

the maximum point is at (-c,d).

Reply 2

Christophicus

xPunkx

Example
y = x^2 + 4x + 6
y = (x+2)^2 -(2)^2 + 6
y = (x+2)^2 + 2

Therefore minimum occurs at (-2,2) since for minimum (x+2)^2 = 0 (since it is squared it is always =/> 0) => x = -2
Plugging x = -2 into the equation gives y = 2

Generally, if you have a quadratic equation;
y = Ax^2 + Bx + C
y = A(x+B/2A)^2 - (B/2A)^2 + C (completing the square)

The quadratic has a local min/max point at (-B/2A, C-B^2/4A^2)

Taking our example, A = 1, B = 4, C = 6
local min point at (-4/2,6-(4^2)/4(1)^2) = (-2,2)