Arithmetic
Quadratic Formula Calculator
Enter the coefficients a, b and c of ax² + bx + c = 0. Get both roots — real or complex — the discriminant, the vertex, the factored and vertex forms, and a live graph of the parabola, all updating as you type.
Enter the coefficients
ax² + bx + c = 0x² − 3x + 2 = 0
Roots
x₁ = 2, x₂ = 1
Two distinct real roots
discriminant
1
vertex x
1.5
vertex y
−0.25
factored: (x − 2)(x − 1) = 0
vertex form: (x − 1.5)² − 0.25
y = 1x² + −3x + 2
parabola roots (x-intercepts) vertex
Try an example
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The basics
What is the quadratic formula?
A quadratic equation is any equation you can write as ax² + bx + c = 0 with a ≠ 0. The quadratic formula is the universal solver: plug in the three coefficients and it returns both roots, every time, whether they are whole numbers, irrational, or complex. The expression under the square root — the discriminant — tells you which kind of answer to expect before you even finish.
x = (−b ± √(b² − 4ac)) ÷ (2a)
The formula itself
The ± gives the two roots: the plus branch and the minus branch. Every other method — factoring, completing the square — agrees with what this returns.
a = 1, b = −3, c = 2 → x = 2 or x = 1
D = b² − 4ac
The discriminant
The quantity under the root. Its sign alone decides whether you get two real roots, one repeated root, or a complex conjugate pair.
D = (−3)² − 4·1·2 = 1 → two real roots
Method
How to solve one by hand
Four steps take you from coefficients to roots — the same path the calculator follows.
- 1
Write the equation in standard form
Rearrange so everything is on one side and zero on the other, then read off a, b and c.
2x² − 7x + 3 = 0 → a = 2, b = −7, c = 3 - 2
Compute the discriminant
Evaluate D = b² − 4ac. Its sign already tells you how many real roots there will be.
D = (−7)² − 4·2·3 = 49 − 24 = 25 - 3
Substitute into the formula
Drop −b, √D and 2a into x = (−b ± √D) ÷ (2a).
x = (7 ± √25) ÷ 4 = (7 ± 5) ÷ 4 - 4
Take both branches
Work the plus and the minus separately to get the two roots, then simplify.
x = 3 or x = ½
Worked examples
The three discriminant cases
Every quadratic falls into one of three families, set by the sign of the discriminant. Switch tabs to see each solved the same way.
Two distinct real roots — the parabola crosses the x-axis twice.
Equation: x² − 5x + 6 = 0
Discriminant: D = (−5)² − 4(1)(6) = 1
Substitute: x = (−(−5) ± √1) ÷ 2 = (5 ± 1) ÷ 2
Two branches: x₁ = 3, x₂ = 2
x = 3 or 2
Reference
What the discriminant tells you
The sign of D = b² − 4ac is a complete preview of the answer — and of how the parabola sits against the x-axis.
D > 0
Two real roots
The square root is a real number, so the plus and minus branches give two different answers. The parabola crosses the x-axis at two points.
D = 0
One repeated root
The square root is zero, so both branches collapse to a single value. The parabola touches the x-axis at exactly one point — its vertex.
D < 0
Two complex roots
The square root is imaginary, so the roots are a conjugate pair p ± qi. The parabola floats entirely above or below the x-axis and never meets it.
Where it shows up
Why quadratics matter
The quadratic is the simplest curved relationship, so it models anything that accelerates, arcs, or trades off against itself.
h = −½g t² + v₀t + h₀
Projectile motion
The height of anything thrown or launched is a quadratic in time. Its roots are the moments it is at ground level.
A = w(L − w)
Area & geometry
Maximizing a rectangular area for a fixed perimeter, or finding a dimension from an area, lands on a quadratic.
P = −q² + bq − c
Business & optimization
Revenue and profit curves are often quadratic in price or quantity; the vertex marks the optimum and the roots mark break-even.
Worth knowing
Rules that make quadratics easy
A handful of relationships turn the formula into something you can sanity-check in your head.
r₁ + r₂ = −b ÷ a
Sum of roots
The two roots add up to −b/a. A fast check: add your answers and compare.
r₁ · r₂ = c ÷ a
Product of roots
The two roots multiply to c/a. Multiply your answers to confirm them — this is Vieta's second formula.
x = −b ÷ (2a)
Axis of symmetry
The vertex and the axis of symmetry sit exactly halfway between the roots, at −b/(2a).
a(x − h)² + k
Vertex form
Completing the square rewrites the quadratic around its vertex (h, k), making the minimum or maximum obvious.
a(x − r₁)(x − r₂)
Factored form
When the roots are real, the quadratic factors over them — the reverse of solving by factoring.
a flips the parabola
The sign of a
If a > 0 the parabola opens upward with a minimum; if a < 0 it opens downward with a maximum.
Pitfalls
Common mistakes to avoid
Most wrong answers come from a small number of sign and bracket slips. Watch for these.
−b, not b
Forgetting to negate b
The formula starts with −b. When b is already negative, −b becomes positive — a sign flip people miss.
(−b)², not −b²
Squaring b correctly
In the discriminant you square b itself, so the result is always non-negative. −b² would give the wrong sign.
÷ 2a, not ÷ 2
Dividing by all of 2a
The whole numerator is divided by 2a — both the 2 and the a. Dividing by 2 alone is a frequent error when a ≠ 1.
standard form first
Not rearranging to = 0
Read off a, b and c only after moving everything to one side. Coefficients from an un-rearranged equation give nonsense.
Quick chart
Discriminant and roots at a glance
Common quadratics with their discriminant, type, and exact roots — a quick reference for checking your own working.
Equation
D = b²−4ac
Type
Roots
x² − 3x + 2 = 0
1
2 real
2, 1
x² − 2x + 1 = 0
0
1 repeated
1 (double)
x² + x + 1 = 0
−3
2 complex
−0.5 ± 0.866i
x² − 4 = 0
16
2 real
2, −2
x² − 5x + 6 = 0
1
2 real
3, 2
x² + 4x + 4 = 0
0
1 repeated
−2 (double)
2x² − 7x + 3 = 0
25
2 real
3, 0.5
x² + 9 = 0
−36
2 complex
± 3i
x² + 6x + 9 = 0
0
1 repeated
−3 (double)
3x² + 2x − 1 = 0
16
2 real
0.3333, −1
x² − x − 6 = 0
25
2 real
3, −2
x² + 2x + 5 = 0
−16
2 complex
−1 ± 2i
FAQ
Common questions
The quadratic formula solves any equation of the form ax² + bx + c = 0, where a is not zero. It states that x = (−b ± √(b² − 4ac)) ÷ (2a). The ± sign produces the two roots — one with the plus branch, one with the minus.
The discriminant is the part under the square root, D = b² − 4ac. Its sign decides everything before you finish: if D > 0 there are two distinct real roots, if D = 0 there is one repeated real root, and if D < 0 there are two complex conjugate roots. You can read the type of answer straight off the sign.
When the discriminant is negative the square root is imaginary, so the equation has no real solutions. The roots come as a conjugate pair p ± qi, where p = −b ÷ (2a) and q = √(−D) ÷ (2a). Graphically, the parabola never crosses the x-axis.
If a = 0 the equation is not quadratic — the x² term disappears and you are left with the linear equation bx + c = 0, whose single solution is x = −c ÷ b (provided b is not also zero). The quadratic formula cannot be used because it divides by 2a, which would be zero.
Factoring is faster when the roots are simple integers or fractions — for example x² − 5x + 6 factors directly to (x − 2)(x − 3). The quadratic formula always works, even when factoring is awkward or the roots are irrational or complex, so it is the reliable fall-back when a factorization is not obvious.
By Vieta's formulas, the two roots r₁ and r₂ satisfy r₁ + r₂ = −b ÷ a and r₁ · r₂ = c ÷ a. These are handy checks: add your two answers and you should get −b/a, multiply them and you should get c/a.
The vertex sits on the axis of symmetry at x = −b ÷ (2a), exactly halfway between the two roots. Substituting that x back into ax² + bx + c gives the vertex's y-value — the minimum of the parabola when a > 0, or the maximum when a < 0.
Yes. The formula works for any real coefficients, whole numbers, decimals, or fractions. The calculator above accepts decimals directly; for fractions, convert them to decimals first or clear the denominators by multiplying the whole equation through, which does not change the roots.
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Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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