Quick Answer – What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable x, written in the standard form ax² + bx + c = 0 where a ≠ 0. It has at most two solutions called roots, found using the quadratic formula or by factorisation.
Theory & Formula
In algebra, a quadratic equation is any equation that can be rearranged into the standard form:
ax² + bx + c = 0 (a ≠ 0)
Here, x represents an unknown variable, and a, b, and c are real coefficients with a not equal to zero. The solutions are given by the quadratic formula:
x = [−b ± √(b² − 4ac)] / 2a
The expression under the square root, D = b² − 4ac, is called the discriminant. The nature of roots depends entirely on D:
• D > 0 → Two distinct real roots
• D = 0 → Two equal real roots (one repeated root)
• D < 0 → Two complex conjugate roots
If a = 0, the equation degenerates to a linear equation bx + c = 0, which has a single root x = −c/b (provided b ≠ 0).
Solved Examples
Example 1: Two Distinct Real Roots
Solve: x² − 5x + 6 = 0
a = 1, b = −5, c = 6
Discriminant D = (−5)² − 4(1)(6) = 25 − 24 = 1
Since D > 0, roots are real and distinct.
x = [−(−5) ± √1] / 2(1) = [5 ± 1] / 2
x₁ = (5+1)/2 = 3, x₂ = (5−1)/2 = 2
Example 2: Equal Real Roots
Solve: x² − 6x + 9 = 0
a = 1, b = −6, c = 9
Discriminant D = (−6)² − 4(1)(9) = 36 − 36 = 0
Since D = 0, roots are real and equal.
x = [−(−6) ± √0] / 2(1) = [6 ± 0] / 2 = 6/2
x₁ = x₂ = 3 (repeated root)
Example 3: Complex Conjugate Roots
Solve: x² + 4x + 13 = 0
a = 1, b = 4, c = 13
Discriminant D = (4)² − 4(1)(13) = 16 − 52 = −36
Since D < 0, roots are complex conjugates.
Real part = −b/(2a) = −4/2 = −2
Imaginary part = √(−D)/(2a) = √36/2 = 6/2 = 3
x₁ = −2 + 3i, x₂ = −2 − 3i
Frequently Asked Questions
Q1. How do I enter negative coefficients in the calculator?
Simply type a minus sign (−) before the number. For example, enter −5 for b if your equation is x² − 5x + 6 = 0. The calculator automatically handles the sign during substitution.
Q2. Can I use decimal values like 0.5 or 1.75?
Yes, absolutely. The calculator accepts decimal numbers with a dot (e.g., 0.5, 3.14, −2.75). You may also use commas (e.g., 0,5) and they will be automatically converted to dots. For fractions like 1/2, please convert to decimal form (0.5) before entering.
Q3. What happens if I enter a = 0?
If a = 0 and b ≠ 0, the equation becomes linear (bx + c = 0) and the calculator solves it accordingly, giving a single real root x = −c/b. If both a = 0 and b = 0, the tool checks whether c = 0 (infinite solutions) or c ≠ 0 (no solution).
Q4. What is the discriminant and why does it matter?
The discriminant D = b² − 4ac determines the nature of the roots without solving the entire equation. If D > 0, there are two distinct real roots; if D = 0, one repeated real root; and if D < 0, two complex conjugate roots. It is a powerful shortcut for JEE and NCERT problems.
Q5. Can this calculator handle scientific notation or very large numbers?
Yes. You can enter numbers in scientific notation such as 1e6 (for 1,000,000) or 2.5e-3 (for 0.0025). The tool gracefully parses these inputs. Extremely large values (beyond 1e308) may cause overflow, but this is rare in standard academic usage.