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The quadratic formula: derivation and use
The formula is derived by completing the square on ax^2 + bx + c = 0. Divide by a, move c/a to the right, add (b/2a)^2 to both sides, take the square root, and solve for x. The result is x = (-b +/- sqrt(b^2 - 4ac)) / 2a.
The formula always works for any quadratic with real coefficients. When the discriminant is negative, it returns complex numbers. Complex roots come in conjugate pairs: if one root is p + qi, the other is p - qi.
The discriminant tells the story
Before computing roots, the discriminant (b^2 - 4ac) tells you what kind of roots to expect. Positive: two distinct real roots (parabola crosses x-axis twice). Zero: one repeated root (tangent to x-axis). Negative: two complex roots (no x-intercepts).
In physics, a negative discriminant often means no physical solution — for example, a projectile that cannot reach a given height. The sign of the discriminant is often more informative than the roots themselves.
Frequently asked questions
- What is the quadratic formula?
- x = (-b plus or minus sqrt(b^2 - 4ac)) divided by (2a), where a, b, c are coefficients of ax^2 + bx + c = 0. The discriminant (b^2 - 4ac) determines how many real roots exist.
- What is the discriminant?
- The discriminant is b^2 - 4ac. If positive: two distinct real roots. If zero: one repeated real root (parabola tangent to x-axis). If negative: two complex conjugate roots (parabola does not cross x-axis).
- What is the vertex of a parabola?
- The vertex is at x = -b/(2a) and y = c - b^2/(4a). If a > 0, the parabola opens upward and the vertex is the minimum. If a < 0, it opens downward and the vertex is the maximum.
- Can I solve a quadratic if a = 0?
- If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula is undefined when a = 0. A linear equation with b not equal to 0 has one solution: x = -c/b.