2026-08-07 10 min

Quadratic Formula Calculator

Solve ax² + bx + c = 0 with the quadratic formula: discriminant, real or complex roots, vertex, and axis of symmetry with steps.

Calculator

Coefficient a

Coefficient b

Constant c

1x² + 5x + 6 = 0

Fractions like 3/4 allowed · results update live

Results update live as you type

Status:Δ = 1x₁ = -2x₂ = -3

Quadratic Solution Summary

Equation1x² + 5x + 6 = 0

Discriminant5² - 4(1)(6) = 1

Rootsx₁ = -2 · x₂ = -3

Vertex / Axis(-2.5, -0.25) · x = -2.5

Results

Normal

Root x₁

-2

(−b + √Δ) / 2a

Root x₂

-3

(−b − √Δ) / 2a

Discriminant Δ

5² - 4(1)(6) = 1

Nature

Two distinct real roots

Root classification from the discriminant sign

Vertex

(-2.5, -0.25)

Turning point (−b/2a, c − b²/4a)

Axis of Symmetry

x = -2.5

Vertical mirror line through the vertex

Roots compute in double precision and display 10 significant digits. Fractions stay exact until evaluation. a = 0 is solved as a linear equation.

Visualizations

Coefficient Magnitudes

1|Δ|

Coefficient Share

a8%

b42%

c50%

Discriminant Sign

Two real100%

Roots on the Axis

x₁2

Formula

Completing the square on ax² + bx + c = 0 isolates x and produces the universal solution. The discriminant under the radical decides whether the two branches land on the real line or in the complex plane.

x = (−b ± √(b² − 4ac)) / (2a)

Alternative Form

Vertex: h = −b/(2a), k = c − b²/(4a); axis x = h

Variables

Symbol	Name	Unit	Description
a	Quadratic coefficient	number	Scales x²; sign sets opening direction
b	Linear coefficient	number	Scales x; shifts vertex horizontally
c	Constant	number	y-intercept of the parabola
Δ	Discriminant	number	b² − 4ac; classifies the roots
h, k	Vertex	point	(−b/2a, c − b²/4a)

Explanation

The ± is the whole story: one formula, two symmetric branches around the axis. When Δ is negative the symmetry continues into complex conjugates.

Derivation

ax² + bx = −c → x² + (b/a)x = −c/a → add (b/2a)² → (x + b/2a)² = (b² − 4ac)/4a² → x = (−b ± √(b² − 4ac))/2a.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write the equation in standard form and read off a, b, c.

ax² + bx + c = 0

a:1

b:5

c:6

Intermediate: 1x² + 5x + 6 = 0

Result: a=1, b=5, c=6

Step 2

Compute the discriminant to classify the roots.

Δ = b² − 4ac

Δ:1

Intermediate: 5² - 4(1)(6) = 1

Result: Δ = 1 (> 0: two real)

Step 3

Take the square root of the discriminant (real or imaginary branch).

√Δ

√Δ:1

Intermediate: √1 = 1

Result: √Δ = 1

Step 4

Apply the full formula for both ± branches.

x = (−b ± √Δ) / 2a

−b:-5

2a:2

Intermediate: (−(5) ± 1) / 2

Result: x₁ = -2, x₂ = -3

Step 5

Locate the vertex and axis of symmetry (skipped for the linear a = 0 case).

h = −b/2a, k = c − b²/4a

h:-2.5

k:-0.25

Intermediate: h = -2.5, k = -0.25

Result: Vertex (-2.5, -0.25), axis x = -2.5

Step 6

Verify by substitution or Vieta: sum = −b/a, product = c/a.

r₁ + r₂ = −b/a; r₁·r₂ = c/a

−b/a:-5

c/a:6

Intermediate: Roots: -2, -3

Result: Substitution returns 0 for each root (within rounding)

Final Answer

-2

Worked Examples

See how the calculator works with different scenarios.

Scenario

Solving x² + 5x + 6 = 0.

Inputs

a:1

b:5

c:6

Calculation

Δ = 25 − 24 = 1 x = (−5 ± 1)/2 x₁ = −2, x₂ = −3

Result

x₁ = −2, x₂ = −3; vertex (−2.5, −0.25)

Interpretation

Positive Δ cuts the axis twice. Vieta checks: sum −5 = −b/a, product 6 = c/a.

Understanding Your Result

Your Result: Root x₁: -2

Typical range: Δ > 0 is the common homework case; Δ = 0 marks perfect squares; Δ < 0 appears in oscillations and non-intersecting geometry.

What This Means

The roots are the x-values zeroing the polynomial; the vertex is its turning point and the axis its mirror line.

Why It Matters

Intersections, optima, and flight times all reduce to finding where a quadratic equals zero or peaks.

Professional Interpretation

Read Δ first: it tells you how many real answers exist before any division happens, and the vertex tells you the extreme value.

Recommendations

•Check Δ before computing roots
•Verify with Vieta sum and product
•Sketch vertex + intercepts to sanity-check signs
•Keep fractions exact until the final display
•Treat a = 0 as linear, not an error

Educational Content

Introduction

Quadratics model anything with constant acceleration or a single optimum: thrown balls, areas with fixed perimeter, revenue with linear demand.

Concept Explanation

Degree 2 forces a parabolic graph with one turning point. Completing the square rewrites any quadratic in vertex form, and the quadratic formula is that rewrite solved for zero.

Applications

▸Projectile and free-fall timing

▸Area and perimeter optimization

▸Braking distance vs speed

▸Profit/revenue maximization

▸Parabolic reflectors and arches

▸Lens and mirror equations

Benefits

•Solves every quadratic, no factoring luck needed
•Δ predicts root type in one line
•Vertex form falls out of the same numbers
•Complex branch keeps algebra closed
•Fraction inputs stay exact until display

Limitations

•Covers single-variable degree 2 only
•Coefficients assumed exact; noisy data needs fitting first
•Display rounds irrationals to 10 digits
•Graph window is illustrative, not a plotter

Best Practices

•Put the equation in standard form first
•Simplify fractions before decimalizing
•Confirm roots by substitution
•Name the vertex when asked for max/min
•State complex pairs as conjugates

Common Misconceptions

Myth: Every quadratic has two real answers.

Truth: Only Δ ≥ 0 gives real roots; negative Δ gives a complex pair.

Myth: a can be zero in a quadratic.

Truth: a = 0 makes it linear — one root, no parabola, no vertex.

Myth: The formula is overkill; factoring always works.

Truth: Factoring needs rational roots; irrationals and complex cases require the formula.

Myth: Vertex and roots are the same thing.

Truth: Roots are x-intercepts; the vertex is the peak/valley — they coincide only for a double root.

Practical Tips

•Read Δ first to know what to expect
•Accept fractions like 3/4 for exact entry
•Use Vieta to check your roots in seconds
•Vertex x = −b/2a doubles as the ± midpoint
•Substitute roots back to verify zero
•Sketch axis + vertex before plotting points
•Re-run after any coefficient change

Common Mistakes

Sign errors on −b

Negate the whole b including its sign: for b = −6, −b = +6.

Forgetting to divide by 2a (dividing by 2 only)

The denominator is the full 2a — with a = 2 it is 4, not 2.

Taking √ of a negative as zero

Negative Δ means factor i out: √(−3) = i√3, giving conjugates.

Leaving the equation non-standard

Move everything left first so c is the true constant term.

Rounding mid-way

Carry √Δ exact until the final division, then display-round.

Assumptions

This calculator makes the following assumptions to provide estimates:

Standard Form: Inputs are the a, b, c of ax² + bx + c = 0.

Real Coefficients: a, b, c are real numbers or real fractions.

Exact Arithmetic: Δ and roots compute in doubles; display rounds to 10 digits.

Linear Fallback: a = 0 is solved as bx + c = 0 when b ≠ 0.

Principal Root: √ denotes the non-negative real root for Δ ≥ 0.

Limitations

Be aware of these limitations when using this calculator:

Degree 2 Only: Cubics and higher need their own solvers.

Single Variable: No systems or parameterized families.

Display Rounding: Irrational and long-decimal roots show 10 digits.

Illustrative Graph: The SVG frames vertex and roots; it is not a full plotter.

Frequently Asked Questions

Answers to common questions about this calculator.

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) / (2a). It solves any ax² + bx + c = 0. The ± gives two branches: one with plus, one with minus.

What is a quadratic equation?

A degree-2 polynomial equation ax² + bx + c = 0 with a ≠ 0. Quadratic comes from quadratum, Latin for square, because the leading term is squared.

How do I use the quadratic formula?

Read a, b, c from standard form, compute Δ = b² − 4ac, take its square root, then evaluate (−b + √Δ)/(2a) and (−b − √Δ)/(2a). Fractions like 3/4 are accepted as inputs.

What is the discriminant?

Δ = b² − 4ac, the quantity under the radical. Its sign classifies the roots before you finish the arithmetic.

How does the discriminant classify roots?

Δ > 0: two distinct real roots (parabola cuts the x-axis twice). Δ = 0: one double root (vertex touches the axis). Δ < 0: two complex conjugates (no x-intercept).

What are real roots?

Real-number solutions that appear as x-intercepts on the graph. They occur exactly when Δ ≥ 0.

What are complex roots?

Solutions involving i = √(−1), always in conjugate pairs p ± qi when Δ < 0. They satisfy the equation algebraically without crossing the real axis.

How do I find the vertex?

h = −b/(2a), k = c − b²/(4a). The vertex (h, k) is the minimum when a > 0 and the maximum when a < 0.

What is the axis of symmetry?

The vertical line x = −b/(2a) through the vertex. Mirroring either half across it reproduces the parabola.

What is vertex form?

y = a(x − h)² + k with (h, k) the vertex. It exposes the turning point, opening direction, and stretch at a glance.

How do I complete the square?

Divide by a, add and subtract (b/2a)² inside, factor the trinomial, and simplify constants. The same steps derive the formula itself.

Can a quadratic have exactly one solution?

Yes when Δ = 0 — a double root. Example: x² − 6x + 9 = 0 gives only x = 3, the vertex sitting on the axis.

What if a = 0?

The equation collapses to bx + c = 0, a linear equation with x = −c/b. This tool solves that case directly and marks vertex fields N/A.

How do I factor a quadratic?

Write it as a(x − r₁)(x − r₂) = 0 using the roots. With a = 1, find two numbers multiplying to c and adding to b — e.g. x² + 5x + 6 = (x + 2)(x + 3).

What are Vieta's formulas?

Sum r₁ + r₂ = −b/a and product r₁·r₂ = c/a. They check solutions fast: for 2x² − 8x + 6 = 0, sum 4 and product 3 confirm roots 1 and 3.

How do I graph a quadratic?

Plot the vertex, draw the axis, mark the y-intercept (0, c) and the roots, then mirror points across the axis and join with a smooth U-curve.

What does the graph look like?

A parabola: U-shaped, opening up for a > 0 and down for a < 0, symmetric about its axis, narrower as |a| grows.

Standard form vs vertex vs factored form?

Standard ax² + bx + c feeds the formula; vertex a(x − h)² + k exposes the turning point; factored a(x − r₁)(x − r₂) exposes the roots. All describe the same curve.

Linear vs quadratic?

Linear mx + b = 0 has degree 1, graphs as a line, one solution. Quadratic has degree 2, graphs as a parabola, up to two solutions.

Does the formula always work?

Yes for every quadratic, real or complex, rational or irrational. Factoring is faster only when integer roots are obvious.

What does a negative discriminant mean geometrically?

The parabola floats entirely above (a > 0) or below (a < 0) the x-axis. Algebraically the two solutions are complex conjugates.

What is an imaginary number?

A multiple of i with i² = −1, arising from square roots of negatives. Combined with reals they form complex numbers a + bi.

Where are quadratics used?

Projectile motion h(t) = −½gt² + v₀t + h₀, area optimization, braking distance (∝ speed²), profit maximization, lens optics, and parabolic arches or dishes.

How do I find max or min values?

Evaluate at the vertex: the extreme value is k = c − b²/(4a), a maximum for a < 0 and minimum for a > 0.

How do I convert to vertex form?

Complete the square: a(x + b/2a)² + (c − b²/4a), so h = −b/(2a) and k = c − b²/(4a).

When can I just take square roots?

When b = 0: ax² + c = 0 rearranges to x = ±√(−c/a). Example: x² − 9 = 0 gives x = ±3.