How the Quadratic Formula Works (Step by Step, With a Worked Example)

The quadratic formula solves any equation of the form ax² + bx + c = 0, and while most people memorize it in school, understanding what each part is actually doing makes it far easier to use correctly — and to know what kind of answer to expect before you even solve it.

The formula itself

Here, a, b, and c are the coefficients from the standard form ax² + bx + c = 0. The ± symbol means the formula produces two solutions (called roots) in most cases — one using +√, one using −√.

The discriminant: knowing the answer type before you solve

The expression under the square root, b² − 4ac, is called the discriminant, and its sign tells you what kind of solutions to expect before doing any further work:

• Positive — two distinct real solutions
• Zero — exactly one real solution (a repeated root)
• Negative — two complex (non-real) solutions

Checking the discriminant first is a useful habit: it catches a common source of errors, since trying to take the square root of a negative number without recognizing it will lead to a wrong or nonsensical real-number answer.

A history of more than 3,500 years

Quadratic problems are among the oldest in mathematics. Babylonian scribes working around 2000 to 1600 BCE solved problems about rectangles with a given area and perimeter that are equivalent to quadratic equations, using geometric methods rather than symbols. In 628 CE the Indian mathematician Brahmagupta gave an explicit rule for solving quadratics in his Brahmasphutasiddhanta, allowing for negative and zero values. Around 820 CE the Persian scholar Muhammad ibn Musa al-Khwarizmi wrote a book on al-jabr, the process of balancing equations, from which the word algebra comes, and described how to solve quadratics by completing the square. The modern notation with letters for coefficients was developed in the sixteenth and seventeenth centuries, and René Descartes' La Géométrie (1637) helped make it standard.

A fully worked example

To solve x² − 5x + 6 = 0, identify a = 1, b = −5 and c = 6. The discriminant is b² − 4ac = 25 − 24 = 1, which is positive, so there are two real roots. Then x = (5 ± √1) ÷ 2, which gives x = 3 and x = 2. You can check by substituting: 3² − 15 + 6 = 0 and 2² − 10 + 6 = 0. This particular equation factors neatly as (x − 2)(x − 3), but the formula works even when factoring is hard to see.

When the roots are not real

For x² + 2x + 5 = 0 the discriminant is 4 − 20 = −16, which is negative. The roots are complex: x = −1 ± 2i. A negative discriminant means the parabola never touches the x-axis. Complex numbers were once regarded with suspicion but are now essential in engineering and physics.

Useful shortcuts from the coefficients

• The sum of the roots equals −b ÷ a, and the product equals c ÷ a. This is known as Vieta's formulas, after the French mathematician François Viète.
• The vertex of the parabola is at x = −b ÷ (2a), which gives the maximum or minimum value.
• If a is positive the parabola opens upward and has a minimum, and if negative it opens downward.

Where quadratics show up

Quadratic equations describe the path of a thrown ball, the area of fenced regions, the stopping distance of vehicles, profit models where price affects sales, and many problems in physics. You can use the Quadratic Formula Calculator on this site to see each step, including the discriminant and the vertex, which is a good way to check your homework or understand where an answer came from.

A full worked example

Solve 2x² + 5x − 3 = 0, where a = 2, b = 5, c = −3.

This gives two solutions: x = (−5 + 7)/4 = 0.5, and x = (−5 − 7)/4 = −3. Plugging either value back into the original equation confirms both satisfy it.

Where the formula comes from

The quadratic formula isn't an arbitrary rule — it's derived by completing the square on the general equation ax² + bx + c = 0. Dividing through by a, moving c/a to the other side, adding (b/2a)² to both sides to form a perfect square trinomial, and then solving for x produces exactly the formula above. This is also why the formula always works, even when factoring by inspection doesn't come easily.