Complex Solutions

Algebra 1 · Axiom Academy

When the discriminant turns negative, a quadratic has no real roots — so we extend the number line with and solve it anyway. Remember the discriminant from the quadratic formula? It decides how many real solutions an equation has: When b^2-4ac < 0 — the discriminant is negative — the whole parabola lifts off the x -axis. Watch the curve rise until it no longer crosses: the two real roots collide, meet, then vanish. 2. When the Discriminant Goes Negative The sign of b^2-4ac is the whole story. Hold a=1 and b=2 fixed and slide c upward: b^2-4ac = 4 - 4c . As c grows the discriminant marches down through zero and into negative territory — and the number of real roots drops . Why does this work? Multiplying by -1 is a turn on the number line ( ). So should be half of that turn — a rotation off the line entirely. Watch a unit length swing up off the real axis: do it twice and you land on -1 . The powers of i cycle every four: a is the real part (left/right on the plane), b is the imaginary part (up/down), and both a and b are ordinary real numbers. A real number only needed one axis. A complex number a+bi needs a whole plane : a steps along the real axis, then b steps along the imaginary axis. Watch each one get plotted from its two coordinates. 5. Solving a Quadratic with Complex Roots

This is the written version of the interactive lesson above. See the full Algebra 1 course.