What is a quadratic equation calculator?

A quadratic equation calculator finds the roots (solutions) of an equation of the form:

ax² + bx + c = 0

where a ≠ 0. The roots are the values of x that make the equation true — geometrically, the points where the parabola y = ax² + bx + c crosses the x-axis.

Quadratic equations have at most two real roots. They may also have one repeated root, or two complex roots (no real solutions). The calculator handles all three cases.

How is it solved?

The standard quadratic formula:

x = (-b ± √(b² − 4ac)) / 2a

The expression under the square root, b² − 4ac, is called the discriminant (Δ). It tells you what type of roots to expect:

Worked example: two real roots

Solve x² − 5x + 6 = 0:

a = 1, b = −5, c = 6
Δ = (−5)² − 4(1)(6) = 25 − 24 = 1
√Δ = 1

x = (5 ± 1) / 2
x₁ = 6 / 2 = 3
x₂ = 4 / 2 = 2

Roots: x = 2, 3

Verify: (2)² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓

Worked example: repeated root

Solve x² − 4x + 4 = 0:

a = 1, b = −4, c = 4
Δ = 16 − 16 = 0
x = (4 ± 0) / 2 = 2

Root: x = 2 (repeated, multiplicity 2)

The parabola touches the x-axis at x = 2 but doesn't cross.

Worked example: complex roots

Solve x² + 2x + 5 = 0:

a = 1, b = 2, c = 5
Δ = 4 − 20 = −16
√Δ = √(−16) = 4i  (where i = √−1)

x = (−2 ± 4i) / 2
x₁ = −1 + 2i
x₂ = −1 − 2i

Roots: complex conjugates

No real solutions — the parabola sits above the x-axis. Complex roots always come in conjugate pairs for real coefficients.

Vertex form

Any quadratic can be rewritten in vertex form:

y = a(x − h)² + k

where (h, k) is the vertex (the minimum or maximum point).

h = −b / 2a
k = c − b²/4a

For x² − 5x + 6:

h = 5/2 = 2.5
k = 6 − 25/4 = −0.25

Vertex: (2.5, −0.25). Below the x-axis (since k < 0), confirming two real roots.

Factored form

If the roots are r₁ and r₂:

ax² + bx + c = a(x − r₁)(x − r₂)

For x² − 5x + 6 = (x − 2)(x − 3) ✓.

Some quadratics factor nicely; many don't. The formula always works.

Vieta's formulas

For roots r₁ and r₂:

Sum:     r₁ + r₂ = −b/a
Product: r₁ × r₂ = c/a

Useful sanity check. For x² − 5x + 6:

Sum: 2 + 3 = 5 ✓ (matches −(−5)/1)
Product: 2 × 3 = 6 ✓ (matches 6/1)

Components and inputs

a, b, c — coefficients

The three numbers from ax² + bx + c = 0. Decimals and negatives allowed. a must not be zero (else it's a linear equation).

Output

• Discriminant (and what it means)
• Real roots (if any), with both values
• Vertex (h, k)
• Factored form (if roots are rational)
• Visual graph (parabola with roots marked)

Common real-world applications

Physics — projectile motion

y(t) = -½gt² + v₀t + y₀

Find when projectile hits ground (y = 0). Solve for t.

Example: a ball thrown up at 20 m/s from 1.5 m height. When does it land?

−4.9t² + 20t + 1.5 = 0
a = −4.9, b = 20, c = 1.5
t = (−20 ± √(400 + 29.4)) / −9.8
  = (−20 ± 20.73) / −9.8
t₁ = (−20 + 20.73)/−9.8 ≈ −0.07 s (before throw, reject)
t₂ = (−20 − 20.73)/−9.8 ≈ 4.16 s ✓

Ball lands at t ≈ 4.16 seconds.

Geometry — area constraints

"A rectangle has perimeter 24 m and area 35 m². Find dimensions."

Length + Width = 12, Length × Width = 35
L(12 − L) = 35
−L² + 12L − 35 = 0
L² − 12L + 35 = 0
L = (12 ± √(144 − 140)) / 2 = (12 ± 2)/2
L = 7 or 5

Dimensions: 7 m × 5 m.

Business — break-even analysis

Profit = Revenue − Cost is quadratic when price affects demand. Setting it to zero gives break-even sales levels.

Algebra homework

"Find x where x² − 7x = −12":

x² − 7x + 12 = 0
a = 1, b = −7, c = 12
Δ = 49 − 48 = 1
x = (7 ± 1)/2 = 4 or 3

Considerations

• Always reduce to standard form first: ax² + bx + c = 0 with everything on the left side.
• Watch the signs: if your equation is 5 − 3x + 2x², that's a = 2, b = −3, c = 5.
• Complex roots aren't "wrong" — they're valid mathematical solutions, just not visible on the real-number x-axis.
• Vertex first, then roots is sometimes easier than the formula — especially for quick sketching.

Limitations

• Solves only quadratics (degree 2). For higher degrees (cubic, quartic), use a polynomial solver.
• Coefficients must be real numbers — doesn't accept complex coefficients in this version.
• Symbolic / algebraic manipulation (factoring symbolically) is not handled — purely numeric.
• Inequalities (x² < 4) not solved — use a separate inequality solver.

Related calculators

• Scientific Calculator — manual computation, includes square root
• Polynomial Solver — higher-degree equations
• Linear Equation — ax + b = 0
• System of Equations — multiple unknowns
• Graphing Calculator — visualize functions
• Distance / Slope — coordinate geometry

Final note. The quadratic formula x = (-b ± √(b²−4ac))/2a is the most useful algebraic tool you'll learn in school. Memorize it once; use it forever. The discriminant tells you the nature of the roots at a glance. For real-world problems (physics, geometry, finance), always check whether negative roots are physical — sometimes you reject one root because it violates the setup (time before zero, negative length, etc.).