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Quadratic Formula Calculator

Math

Solve any quadratic equation instantly. Enter coefficients a, b, and c to find real and complex roots with step-by-step working shown for free.

Reviewed by the thecalcu.com team · Last updated June 21, 2026

x² − 5x + 6 = 0

a(x²)
b(x)
c(const)

Enter ax² + bx + c = 0 coefficients

Discriminant (Δ)

0
Complex roots (Δ < 0)

Roots

x₁ =0
x₂ =0

Complex roots come in conjugate pairs. The equation has no real solutions.

Formula

x = (−b ± √(b²−4ac)) / 2a

Δ = (-5)² − 4(1)(6) = 25 − 24 = 0

What is a Quadratic?

The Quadratic Formula Calculator solves any second-degree polynomial equation of the form ax² + bx + c = 0 instantly, showing the discriminant, both roots, and step-by-step working. A quadratic equation is the mathematical backbone of countless real-world situations, from calculating the dimensions of a rectangular plot of land to modelling the trajectory of a cricket ball or analysing profit-maximisation problems in business.

The equation ax² + bx + c = 0 has three parameters: a (the coefficient of x²), b (the coefficient of x), and c (the constant term). The value of a must be non-zero; if it equals zero, the equation degenerates to a linear equation. The standard quadratic formula x = (−b ± √(b²−4ac)) / 2a derives every possible root algebraically, regardless of whether those roots are whole numbers, irrational surds, or complex numbers.

Quadratic equations feature heavily in the CBSE and ICSE syllabi for Classes 9 through 12, making them one of the most frequently searched mathematics topics in India. Beyond academics, they arise in engineering design, financial modelling, and physics. For example, if you launch a rocket vertically and want to know when it returns to the ground, the height-versus-time relationship is quadratic. The Slope Calculator handles linear relationships (degree 1), while the quadratic formula handles degree-2 problems, together they cover the vast majority of algebraic equations students and professionals encounter.

The discriminant (Δ = b² − 4ac) is the key indicator: positive means two distinct real roots, zero means one repeated real root (the parabola is tangent to the x-axis), and negative means two complex conjugate roots with no real-number solutions. Understanding the discriminant is essential before attempting to factorise, if Δ is not a perfect square, factoring over integers will fail and the formula is the only clean path to the answer.

Why Use a Quadratic Formula Calculator?

Solving quadratic equations by hand is error-prone, particularly when coefficients are large, fractional, or negative. A sign error in computing −4ac or a slip under the square root produces completely wrong roots, yet nothing in the arithmetic warns you. This calculator eliminates those errors by computing the discriminant and roots exactly, displaying every intermediate step so you can verify the working.

The instant feedback loop makes it ideal for "what-if" exploration. Suppose you are designing a rectangular enclosure with a fixed perimeter and want to maximise its area, adjusting the coefficient values lets you see immediately how the roots shift and whether they remain real and positive (a requirement for physical dimensions). For students preparing for competitive exams such as JEE Main, which frequently sets quadratic-equation questions requiring root analysis, the step-by-step breakdown reinforces the method rather than just giving an answer.

The calculator also handles the special cases that trip up manual computation: equations with a = 0 (switches to linear mode), discriminant exactly zero (returns the single repeated root), and negative discriminants (returns complex roots in the form p ± qi). Checking your answer against the calculator before submitting an exam paper or engineering report is a low-effort, high-confidence verification step.

Who Should Use This Calculator?

Students in Classes 9–12 working through the CBSE or ICSE mathematics syllabus will encounter quadratic equations in nearly every chapter of algebra and coordinate geometry. The step-by-step breakdown helps students understand why each operation is performed, not just what the answer is.

JEE and competitive exam aspirants often face quadratic equations with large or irrational coefficients where manual calculation is slow and error-prone. Using the calculator to verify answers during practice allows aspirants to focus mental energy on problem setup rather than arithmetic.

Engineers and physicists who need to solve second-degree characteristic equations in dynamics, circuit analysis, or control systems will find the discriminant indicator useful for quickly assessing solution type before deeper analysis.

Business analysts and economists dealing with revenue-versus-cost breakeven models, pricing optimisation, or quadratic cost functions can use this tool to identify breakeven quantities or profit-maximising output levels without setting up the formula from scratch each time.

Mathematics teachers and tutors can use the calculator to generate worked examples with specific root types (irrational, complex, repeated) for classroom demonstration, using the Area Calculator alongside it when geometry-based word problems require both tools.

What Insights Does the Quadratic Calculator Give You?

Discriminant (Δ), the primary diagnostic output. A positive Δ confirms two real solutions exist; the magnitude tells you how far apart the roots are. A zero Δ signals a perfect-square quadratic. A negative Δ means the problem has no real-number answer, which in physics indicates an impossible scenario (e.g., the projectile never reaches that altitude) and in engineering indicates oscillatory underdamped behaviour.

Root 1 (x₁), the larger of the two real roots when Δ > 0, computed as (−b + √Δ) / 2a. In physical problems, this is often the primary answer you need, for instance, the later time at which a falling object hits the ground.

Root 2 (x₂), the smaller real root, computed as (−b − √Δ) / 2a. In word problems, this root may be inadmissible (negative value for a length or time) and should be checked against the problem's constraints before accepting.

When roots are complex (Δ < 0), the calculator displays them in the form p ± qi, where p = −b/2a is the real part and q = √|Δ|/2a is the imaginary part. The sum of the two roots always equals −b/a and their product always equals c/a, these Vieta's formulae are useful cross-checks. Pair this tool with the Standard Deviation Calculator when working with datasets where quadratic regression is involved.

How to use this Quadratic calculator

  1. Write your equation in standard form, rearrange any quadratic expression so all terms are on one side: ax² + bx + c = 0. For example, 2x² = 7x − 3 becomes 2x² − 7x + 3 = 0, giving a = 2, b = −7, c = 3.

  2. Enter Coefficient a (x²), type the coefficient of the squared term. Include the sign; if the x² term is negative (e.g., −3x²), enter −3. If the coefficient is 1 (just "x²"), enter 1.

  3. Enter Coefficient b (x), type the coefficient of the linear term. If there is no x term, enter 0. Remember to include the sign from the original equation.

  4. Enter Coefficient c (constant), type the constant term. If there is no constant, enter 0.

  5. Read the discriminant, the calculator displays Δ and labels the root type (two real roots / one real root / complex roots) with a colour indicator so you immediately know what kind of solution to expect.

  6. Read the roots, if Δ ≥ 0, both roots are shown as decimals. If Δ < 0, roots are displayed in complex form. Use the step-by-step breakdown below the result to follow the calculation for exam revision.

  7. Verify with Vieta's formulae, mentally check that x₁ + x₂ = −b/a and x₁ × x₂ = c/a as a sanity check on both values.

Formula & Methodology

The Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

Variable definitions:
- a, coefficient of x² (must not be zero)
- b, coefficient of x (can be any real number, including zero)
- c, constant term (can be any real number, including zero)
- Δ, discriminant = b² − 4ac
- √Δ, square root of discriminant (real if Δ ≥ 0, imaginary if Δ < 0)

Root expressions:
- x₁ = (−b + √Δ) / 2a
- x₂ = (−b − √Δ) / 2a

Vieta's formulae (verification):
- Sum of roots: x₁ + x₂ = −b / a
- Product of roots: x₁ × x₂ = c / a

Worked example, fencing a rectangular field:

A farmer wants to fence a rectangular plot where the length exceeds the width by 8 metres and the total area is 240 sq m. Let width = x metres. Then length = x + 8, and:

x(x + 8) = 240x² + 8x − 240 = 0

So a = 1, b = 8, c = −240.

Δ = b² − 4ac = 8² − 4(1)(−240) = 64 + 960 = 1024

√Δ = √1024 = 32

x₁ = (−8 + 32) / 2 = 24/2 = 12 metresx₂ = (−8 − 32) / 2 = −40/2 = −20 metres (inadmissible, negative width)

The width is 12 m and the length is 20 m. Verification: 12 × 20 = 240 sq m ✓. The Area Calculator can confirm the rectangle's area independently.

Assumption: The formula assumes the equation has been reduced to exact rational or irrational coefficients. For equations derived from physical measurements, round-off in coefficients will propagate to round-off in roots; the step-by-step display helps quantify this sensitivity.

Frequently Asked Questions

The quadratic formula is x = (−b ± √(b²−4ac)) / 2a, used to find the roots of any quadratic equation in the form ax² + bx + c = 0. It works by completing the square algebraically on the general form, deriving a universal solution. The ± sign produces two possible roots, which may be real and distinct, real and equal, or complex depending on the discriminant.
The discriminant is the expression Δ = b² − 4ac found under the square root in the quadratic formula. It determines how many real solutions the equation has: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one repeated real root; if Δ < 0, the roots are complex conjugates with no real solutions. The discriminant is the single most important diagnostic value in quadratic analysis.
When Δ < 0, the quadratic equation has no real roots, instead, it has two complex conjugate roots of the form p ± qi, where i = √(−1). This means the parabola represented by y = ax² + bx + c does not cross the x-axis at any real point. Complex roots appear frequently in electrical engineering, signal processing, and quantum mechanics, but in most school-level problems a negative discriminant indicates no real solution exists.
Enter the three coefficients a, b, and c from your equation ax² + bx + c = 0 into the respective input fields. The calculator instantly computes the discriminant, displays whether roots are real or complex, and shows both root values with full step-by-step working. You can also use it in reverse: rearrange any quadratic expression into standard form first, then input the coefficients.
A linear equation has the highest power of the variable as 1 (e.g., 2x + 3 = 0), yielding exactly one solution. A quadratic equation has the highest power as 2 (e.g., x² − 5x + 6 = 0) and can yield zero, one, or two solutions. Linear equations are solved by simple rearrangement, while quadratic equations require the quadratic formula, factoring, or completing the square.
A repeated root (also called a double root) occurs when the discriminant equals zero. Both roots have the same value: x = −b / 2a. Geometrically, this means the parabola y = ax² + bx + c touches the x-axis at exactly one point (the vertex) without crossing it. In practice, repeated roots arise when a quadratic is a perfect square, such as (x − 3)² = 0.
You can solve by factoring (if the equation factors neatly), by completing the square, or by directly applying the quadratic formula x = (−b ± √(b²−4ac)) / 2a. Factoring works well for simple integer roots but fails for irrational or complex roots. The quadratic formula always works regardless of the nature of the roots, making it the most reliable manual method for all cases.
Quadratic equations appear in land and construction problems (finding dimensions of a plot with a given area), projectile motion in physics (height of a ball thrown upward), break-even analysis in business (where revenue equals cost), and optimisation problems in engineering and economics. They are a core topic in the CBSE and ICSE Class 10 and Class 11 mathematics syllabi.
No, if a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula is undefined when a = 0 because dividing by 2a becomes division by zero. In that case, the single solution is simply x = −c / b, provided b ≠ 0. Our calculator handles this edge case by switching to linear mode automatically.
The two roots x₁ and x₂ of ax² + bx + c = 0 satisfy Vieta's formulae: their sum x₁ + x₂ = −b/a and their product x₁ × x₂ = c/a. When the discriminant is positive, the difference between roots is √Δ / |a|. The two roots may be integers, fractions, surds, or complex numbers depending on the values of a, b, and c.
Yes, cubic equations (degree 3) have Cardano's formula, and quartic equations (degree 4) have Ferrari's formula, but both are vastly more complex than the quadratic formula. For degree 5 and above, there is no general algebraic formula (Abel–Ruffini theorem), and numerical methods are used instead. For most practical and educational purposes, quadratic equations are the highest degree solved analytically.
Identify the unknown quantity and call it x. Write an algebraic expression for the given relationships, for example, if the product of two consecutive integers equals 72, write x(x+1) = 72, which gives x² + x − 72 = 0. Rearrange to standard form ax² + bx + c = 0, identify the coefficients, and apply the quadratic formula. Always check both roots against the original problem context, as negative or non-integer roots may be inadmissible.
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