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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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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)
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Perfect Mirror!

Reverse Engineering

Given the conjugate roots , build the original equation.
Hint: Sum = , Product =

+
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Master of the Complex!

You've successfully navigated the imaginary plane, tamed the discriminant, and built quadratics from scratch.