Solve any quadratic equation using the quadratic formula, factoring, or completing the square — real and complex roots.
Δ = b² − 4ac
Δ > 0 — Two distinct real roots. The parabola crosses the x-axis at two points.
Δ = 0 — One repeated real root (double root). The parabola is tangent to the x-axis at the vertex.
Δ < 0 — Two complex conjugate roots (a ± bi). The parabola does not cross the x-axis at all.
x₁ + x₂ = −b/a x₁ · x₂ = c/a
Vieta's formulas let you verify roots without substituting back. If x₁ and x₂ are your two roots:
Their sum must equal −b/a, and their product must equal c/a.
This is a fast mental check — especially handy when you factor by inspection.
One-on-one Algebra tutoring builds real intuition for when to factor, when to complete the square, and when to just use the formula — we work through your actual homework and tests.