polypad art mode

A Quadratic, Seen All At Once

Move a coefficient. The whole proof changes.
Interactive quadratic art muralvertex12Equationx^2 - 3x + 2 = 0D = 1 | x = 1, 2Complete the square(x - 1.5)^2 = 0.25x = 1.5 +/- 0.5p = -3 | q = 2
parameter atlasD = b^2 - 4ac

The current point lives in coefficient space. Blue regions have two real roots, gold is the tangent boundary, and red is the complex-root region.

Discriminant map in b c coefficient spacecb
root planeReal-axis roots

Real roots sit on the horizontal axis. Complex roots rise into mirrored positions above and below it.

Roots shown on the real and imaginary axes12realimaginary
standardx^2 - 3x + 2 = 0
factor(x - 1)(x - 2)
vertexy = 1(x - 1.5)^2 - 0.25
square(x - 1.5)^2 = 0.25
Discriminant lens1

Two visible crossings

x2
-1.5x
-1.5x
2.25
balanced area0.25
Area becomes side lengthx = 1.5 +/- 0.5
Why this is a polypadEvery visual object is tied to the same equation.

The curve shows roots and vertex, the sculpture shows completing the square, and the discriminant lens explains whether the solutions appear on the real axis or as a complex pair.

Current solutionx = 1, 2

(x - 1.5)^2 = 0.25