The quadratic formula solves any equation of the form:
The solutions — the values of that make the equation true — are given by:
The symbol means there are generally two solutions, one using and one using .
Why it works
The derivation uses a technique called completing the square. Start with , divide through by , move the constant, and rewrite the quadratic term as a perfect square. The result, after algebraic simplification, is the formula above.
The quantity under the square root, , is called the discriminant. It tells you what kind of solutions to expect:
- Positive discriminant → two distinct real solutions
- Zero discriminant → one repeated real solution (a “double root”)
- Negative discriminant → two complex solutions (conjugates)
Geometrically, the solutions are the points where the parabola crosses the -axis. A positive discriminant means the parabola crosses twice, zero means it just touches, negative means it never crosses at all.
History
Solving quadratics is one of the oldest mathematical activities. Babylonian mathematicians around 2000 BCE had procedures for specific cases — they used geometric arguments and tables of numerical values. Ancient Greek mathematicians reformulated the problem geometrically, and medieval Indian mathematicians (Brahmagupta, 7th century) wrote down a general algebraic rule.
The credit for systematizing the solution usually goes to the Persian mathematician Muhammad ibn Musa al-Khwarizmi, whose 9th-century work Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala gave us the word “algebra” (from al-jabr, “restoration”). Al-Khwarizmi classified quadratic equations and gave step-by-step solution procedures for each type.
The modern symbolic form of the formula, with letters standing for general coefficients, emerged during the European Renaissance, culminating in work by Cardano, Tartaglia, and Vieta in the 16th century.
What comes next
Cubic equations () and quartic equations (degree 4) also have general solution formulas, discovered by Italian mathematicians in the 16th century. But quintics — degree-5 polynomials — do not have a general formula using only roots and basic operations. This was proved independently by Niels Henrik Abel and Évariste Galois in the early 19th century. Their proof gave birth to Galois theory, one of the most beautiful parts of modern algebra.
Why students remember it
In many parts of the world the quadratic formula is the first formula students memorize that genuinely solves a whole class of problems in one stroke. It is simple enough to commit to memory, powerful enough to matter, and it connects a symbolic manipulation (the formula) to a geometric picture (the parabola) in a way that shows what “understanding mathematics” really means.
The sign of the discriminant Δ = b² − 4ac determines whether the parabola crosses the x-axis twice, is tangent to it, or misses it entirely.
Frequently asked
Who invented the quadratic formula?
No single person. Babylonian mathematicians solved specific quadratic equations around 2000 BCE. Al-Khwarizmi systematized the solution in the 9th century, and the modern symbolic formula emerged during the European Renaissance.
What is the discriminant?
The expression b² − 4ac under the square root. If it is positive, there are two real solutions; if zero, one repeated solution; if negative, two complex solutions.
Does every polynomial have a formula like this?
For degrees 2, 3, and 4 — yes. But Évariste Galois proved in 1832 that no general formula exists for polynomials of degree 5 or higher using only roots and basic operations.