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Quadratic Formula Calculator

Enter the "coefficients (a, b, and c)" or the "full quadratic equation" to calculate roots, discriminant (Δ), and graph with step-by-step calculations.

x²

= 0

Enter your quadratic equation:

Enter equations like: x^2 + 5x + 6 = 0, 2x^2 - 3x + 1 = 0, or x^2 - 4 = 0

Related

Quadratic Formula Calculator

Quickly solve any quadratic equation of the form ax² + bx + c = 0 using this calculator. It calculates real or complex roots, shows the discriminant, and even displays a visual graph of the parabola. Ideal for students, engineers, and anyone needing fast and accurate solutions.

What Is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2 in a single variable.

Standard Form:

ax² + bx + c = 0

Where:

x = variable
a, b, c = real numbers
a ≠ 0, otherwise it is a linear equation

The term “quadratic” comes from the Latin word quadratus, meaning “square.” The solutions of the equation are called roots or zeros. Depending on the discriminant, roots can be distinct, repeated, or complex.

Practical Uses of Quadratic Equations:

Predicting projectile motion or trajectory
Finding maximum or minimum values in optimization problems
Modeling parabolic paths in physics, engineering, or economics

The Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

Where:

a = coefficient of x²
b = coefficient of x
c = constant term
b² - 4ac = discriminant (Δ), which determines the type of roots

Roots Based on Discriminant

Δ > 0 → two distinct real roots
Δ = 0 → one repeated real root
Δ < 0 → two complex conjugate roots

Derivation of the Quadratic Formula

Step-by-step derivation by completing the square:

Start with the general equation: ax² + bx + c = 0
Divide all terms by a: x² + (b/a)x + c/a = 0
Move the constant to the other side: x² + (b/a)x = -c/a
Complete the square: add (b/2a)² to both sides → x² + (b/a)x + (b/2a)² = -c/a + (b²/4a²)
Factor left-hand side: (x + b/2a)² = (b² - 4ac)/4a²
Take square root: x + b/2a = ±√(b² - 4ac)/2a
Solve for x: x = (-b ± √(b² - 4ac)) / 2a

Using the Quadratic Formula Calculator

Option 1: Enter Coefficients

Enter values for a, b, and c
Click “Calculate”
View the discriminant, roots, and parabola graph

Option 2: Enter Full Equation

Input the equation in standard form
Click “Calculate” → the calculator automatically identifies a, b, and c
See the roots, discriminant, and curve of the parabola

Types of Roots

1. Two Real Distinct Roots (Δ > 0)

The parabola crosses the x-axis at two points.

Example: x² - 7x + 12 = 0 → Δ = 1 → x₁ = 3, x₂ = 4

2. One Repeated Root (Δ = 0)

The parabola touches the x-axis at a single point.

Example: x² - 6x + 9 = 0 → Δ = 0 → x = 3

3. Complex Roots (Δ < 0)

The parabola does not intersect the x-axis.

Example: x² + 2x + 5 = 0 → Δ = -16 → x = -1 ± 2i

Real-Life Example

A ball is thrown with an initial height of 2 m and speed 15 m/s. Find when it hits the ground.

Equation: y = -4.9t² + 15t + 2 → a = -4.9, b = 15, c = 2 → y = 0

Discriminant: Δ = b² - 4ac = 281.6 → two real roots
Quadratic formula: t = (-b ± √Δ) / 2a
Compute roots: t₁ ≈ -0.13 s (discarded), t₂ ≈ 6.28 s

The ball hits the ground after ~6.28 seconds.

FAQs

Can I use this calculator if a = 0?

No. The equation becomes linear in that case (bx + c = 0).

Does the calculator show detailed steps?

Yes. It shows discriminant, substitution into the formula, and the roots calculation.

How can I tell if a quadratic has two solutions?

If Δ > 0 → the parabola intersects the x-axis at two points, indicating two real solutions.

Key Points

This calculator makes solving quadratic equations fast and accurate, useful for homework, physics problems, and checking calculations with step-by-step details.

References

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