Concept · Algebra
What Is the Quadratic Formula? Where It Comes From
One formula solves every quadratic there is. It is worth knowing why, because the reason is also the method it replaces.
The formula, and what each part does
The short answer to what is the quadratic formula: it is the general solution of any equation of the form ax² + bx + c = 0, written in terms of the three coefficients.
- ax² + bx + c = 0any quadratic, in standard form
- x = (−b ± √(b² − 4ac)) ÷ 2athe formula that solves it
Three pieces are doing separate jobs. The −b sets where the roots are centred. The ± produces two roots at equal distance either side of that centre. And b² − 4ac, the discriminant, decides how far apart they are — or whether they exist at all.
What the discriminant tells you first
The expression under the root is worth evaluating on its own, before touching anything else, because it settles the shape of the answer in one line.
| Discriminant b² − 4ac | Roots | On the graph |
|---|---|---|
| positive | two distinct real roots | the parabola crosses the x-axis twice |
| zero | one repeated root | it touches the axis at exactly one point |
| negative | no real roots | it never reaches the axis |
A positive discriminant that is also a perfect square means something extra: the roots are rational, so the quadratic factors over the integers and factoring would have been quicker.
Where the formula comes from
It is not a rule handed down from nowhere. It is what completing the square produces when you do it with letters instead of numbers — once, in general, so that nobody has to repeat the work for each equation.
- ax² + bx + c = 0the equation
- x² + (b/a)x = −c/adivide by a, move c across
- (x + b/2a)² = (b² − 4ac)/4a²complete the square on the left
- x + b/2a = ±√(b² − 4ac)/2atake the root of both sides
- x = (−b ± √(b² − 4ac))/2asubtract b/2a — the formula
Seeing the derivation explains the ± too. Taking the square root of both sides of an equation always admits two signs, and that single step is the origin of the two roots.
How it relates to the graph
Every quadratic draws a parabola, and the formula describes it. The centre term −b/2a is exactly the axis of symmetry, so the two roots sit symmetrically either side of the vertex.
That is why the discriminant reads so cleanly off a picture. If the vertex is below the axis and the parabola opens upward, it must cross twice; if the vertex sits on the axis, the two roots have collapsed into one.
When there are no real roots
A negative discriminant does not mean the equation is broken. It means no real number satisfies it, because no real number squares to a negative.
The roots still exist as a complex conjugate pair, written a ± bi where i is defined as √−1. Every quadratic has exactly two roots once complex numbers are allowed — which is the tidier statement, and the reason complex numbers were invented in the first place.
Formula or factoring?
Both give the same roots. The difference is speed and reliability.
- Factoring is faster when it works, but it only works when the discriminant is a perfect square.
- The formula always works, on every quadratic, including the ones with irrational or complex roots.
- Completing the square is slower than both, and worth knowing anyway because it is where the formula came from and it generalises to conic sections.
A practical order: check the discriminant, factor if it is a perfect square, otherwise use the formula.
Questions about what is the quadratic formula
What is the quadratic formula used for?
Finding both roots of any equation of the form ax² + bx + c = 0 — the values of x that make it true. On a graph they are where the parabola crosses the x-axis.
What does b² − 4ac tell you?
It is the discriminant. Positive gives two real roots, zero gives one repeated root, and negative gives none, because no real number has a negative square.
Where does the quadratic formula come from?
From completing the square on ax² + bx + c = 0 using letters rather than numbers. The result is a general solution, so the work never has to be repeated.
Does the formula work on every quadratic?
Yes, provided a is not zero. If a is zero the equation is linear, not quadratic, and it is solved in one step instead.