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What Is Completing the Square

Result

(x + 5)^2 + 3

Plots

Geometric figure

parabola

Expanded form

x^2 + 10 x + 28

Alternate forms

x (x + 10) + 28

(-i x + sqrt(3) - 5 i) (i x + sqrt(3) + 5 i)

Complex roots

x = -5 - i sqrt(3)

x = -5 + i sqrt(3)

Roots in the complex plane

Polynomial discriminant

Δ = -12

Properties as a real function

R (all real numbers)

{y element R : y>=3}

Derivative

d/dx((x + 5)^2 + 3) = 2 (x + 5)

Indefinite integral

integral(3 + (5 + x)^2) dx = x^3/3 + 5 x^2 + 28 x + constant

Global minimum

$min{(x + 5)^2 + 3} = 3 at x = -5$