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If a parabola floats entirely above the horizontal axis, it never touches y=0y = 0 in the real coordinate plane.
Yet the Fundamental Theorem of Algebra proves that every quadratic equation has exactly two solutions — they are just hidden in the complex plane.
To calculate these hidden roots, we need a single algebraic tool that defines the square root of a negative value.

The Imaginary Unit and Standard Form

In 1545, Italian mathematician Girolamo Cardano first published calculations involving square roots of negative numbers while solving cubic equations, calling the process mental torture before realizing its mathematical power.
Today, we define the imaginary unit as i=−1i = \sqrt{-1}, which means that i2=−1i^2 = -1.
A complex number combines a real number and an imaginary number into standard form: a+bia + bi, where aa is the real part and bb is the imaginary part (both aa and bb are real numbers).
What happens when our quadratic formulas produce a negative number underneath the square root symbol?

Solving with the Quadratic Formula

For any standard quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the roots are given by the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The expression inside the radical, b2−4acb^2 - 4ac, is the discriminant; when it is negative, we factor out −1\sqrt{-1} as ii using the rule −N=N⋅−1=iN\sqrt{-N} = \sqrt{N} \cdot \sqrt{-1} = i\sqrt{N} for any positive real number NN.