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The Missing Roots
When a parabola doesn't cross the x-axis, its roots are complex numbers.
Discriminant
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Drag the vertex up
Extracting the Imaginary
Let's solve using the quadratic formula.
Conjugate Pairs
Complex roots always come in pairs mirrored across the Real axis. They are called Complex Conjugates.
Root 1
Root 2 (Target)
Drag the blue point
Perfect Mirror!
Reverse Engineering
Given the conjugate roots , build the original equation.
Hint: Sum = , Product =
+
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+
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Drag numbers to boxes
Master of the Complex!
You've successfully navigated the imaginary plane, tamed the discriminant, and built quadratics from scratch.