Quadratic Calculator

Solves equations of the form: ax² + bx + c = 0

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term that is squared. The standard form is written as:

Where:

• x represents the unknown variable.
• a, b, and c represent known numbers or coefficients.
• Crucially, a cannot be zero (a ≠ 0), otherwise the equation becomes linear, not quadratic.

Solving a quadratic equation means finding the value(s) of x that make the equation true. These values are called the roots or solutions of the equation. Graphically, the real roots of a quadratic equation correspond to the x-intercepts (where the graph crosses the x-axis) of the parabola represented by y = ax² + bx + c.

How This Quadratic Calculator Works

This calculator uses the well-known Quadratic Formula to find the roots (x) of the equation ax² + bx + c = 0. Here's the process:

1. Enter Coefficients: Input the values for the coefficients a, b, and c into the respective fields. Remember that a cannot be zero.
2. Calculate Discriminant: The calculator first computes the discriminant (often denoted by delta, Δ), using the formula: Δ = b² - 4ac The value of the discriminant determines the nature of the roots.
3. Apply Quadratic Formula: Based on the discriminant, the calculator applies the quadratic formula to find the roots: x = [-b ± √( b² - 4ac )] / ( 2a ) Which is the same as: x = (-b ± √Δ) / (2a)
4. Determine Root Type & Display Results:
   • If Δ > 0: There are two distinct real roots (x₁ and x₂).
   • If Δ = 0: There is exactly one real root (a repeated root). The calculator will show the same value for x₁ and x₂.
   • If Δ < 0: There are two complex conjugate roots (involving the imaginary unit i, where i = √-1). The calculator will display these in the form p ± qi.
   The calculator displays the value of the discriminant, the nature of the roots, and the calculated root(s) x₁ and x₂.

Input validation ensures that a is not zero and that all inputs are valid numbers.

Frequently Asked Questions (FAQs)

Q1: What happens if the coefficient 'a' is zero?

If a = 0, the equation is no longer quadratic; it becomes a linear equation (bx + c = 0). This calculator requires a to be non-zero. If you enter 0 for a, it will display an error message. To solve a linear equation, simply rearrange it to x = -c / b (assuming b ≠ 0).

Q2: What does the Discriminant (Δ) tell me?

The discriminant (Δ = b² - 4ac) is a powerful part of the quadratic formula. It tells you about the number and type of roots *without* having to calculate the full roots:

• Δ > 0: Two distinct real roots. The parabola crosses the x-axis at two different points.
• Δ = 0: One real root (or two equal real roots; a repeated root). The vertex of the parabola touches the x-axis at exactly one point.
• Δ < 0: Two complex conjugate roots (no real roots). The parabola does not cross the x-axis.

This calculator explicitly shows the discriminant value and states the nature of the roots based on it.

Q3: What are Complex Roots?

When the discriminant is negative (Δ < 0), the quadratic formula requires taking the square root of a negative number. This leads to roots that involve the imaginary unit i, where i = √-1. These are called complex roots. They always come in conjugate pairs, meaning if p + qi is a root, then p - qi is also a root (where p is the real part and q is the imaginary part). This calculator displays complex roots in the standard p ± qi format.

Q4: Can the coefficients a, b, and c be negative or decimals?

Yes. The calculator accepts positive numbers, negative numbers, and decimals (floating-point numbers) for the coefficients a, b, and c. Just ensure that a is not exactly zero.

Q5: How accurate are the calculated roots?

The calculator uses standard JavaScript floating-point arithmetic (double-precision). The results are generally very accurate for most inputs. The displayed roots are typically rounded to a reasonable number of decimal places for readability. For calculations requiring extreme precision, specialized mathematical software might be needed.

Q6: Why is solving quadratic equations important?

Quadratic equations appear frequently in various fields, including physics (e.g., projectile motion), engineering (e.g., circuit analysis, structural design), economics (e.g., optimization problems), finance, and geometry. Understanding how to solve them is a fundamental skill in mathematics and science.