Quadratic Formula Calculator — Solve ax² + bx + c

x² − 5x + 6 = 0

a(x²)

b(x)

c(const)

Enter ax² + bx + c = 0 coefficients

Discriminant (Δ)

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.

How to use this Quadratic calculator

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

What is the quadratic formula and how does it work?

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.

What is the discriminant in a quadratic equation?

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.

What happens when the discriminant of a quadratic equation is negative?

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.

How do I use the Quadratic Formula Calculator?

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.

What is the difference between a linear equation and a quadratic equation?

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.

What is a repeated root in a quadratic equation?

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.

How to solve a quadratic equation manually without a calculator?

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.

What are real-life applications of quadratic equations in India?

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.

Can the quadratic formula be used if the coefficient a is zero?

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.

What is the difference between the two roots of a quadratic equation?

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.

Is there a formula to solve cubic or higher-degree equations?

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.

How do I convert a word problem into a quadratic equation?

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.