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ax² + bx + c = 0
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How the Quadratic Formula Works
The Quadratic Formula Calculator solves any equation of the form ax² + bx + c = 0 and returns both roots, the discriminant, and the nature of the roots (real and distinct, real and equal, or complex). Enter the coefficients a, b, and c and the result is calculated instantly.
The quadratic formula is one of the most reliably tested topics in Class 10, Class 11, and competitive exams like JEE. It handles every case in a single formula: equations with two different real solutions, equations with exactly one solution, and equations where no real solution exists. This calculator also shows the working, so you can verify each step against your own.
What the Coefficients Mean
Every quadratic equation must be rearranged into the standard form ax² + bx + c = 0 before using the formula. Once in that form, identify:
- a — the coefficient in front of x² (cannot be zero; if a = 0, the equation is linear)
- b — the coefficient in front of x (can be zero, negative, or positive)
- c — the constant term with no x (can be zero, negative, or positive)
For the equation 2x² − 5x + 3 = 0, the values are a = 2, b = −5, c = 3. The negative sign belongs to b, not to x itself. This is a common point of error when reading off coefficients.
Understanding the Discriminant
The discriminant is the expression b² − 4ac, which is calculated first before applying the full quadratic formula. Its value determines the nature of the roots before you even finish the calculation.
| Discriminant Value | Nature of Roots | What It Looks Like on a Graph |
|---|---|---|
| Greater than 0 | Two distinct real roots | The parabola crosses the x-axis at two different points |
| Equal to 0 | One repeated real root | The parabola just touches the x-axis at exactly one point |
| Less than 0 | Two complex (imaginary) roots | The parabola does not cross the x-axis at all |
Checking the discriminant first is good practice in exam conditions because it tells you immediately how many solutions to expect, which helps you catch arithmetic errors before completing the full calculation.
The Formula and a Worked Example
Where:
- a, b, c — the coefficients from the standard form ax² + bx + c = 0
- ± — two solutions are calculated: x&sub1; uses addition, x&sub2; uses subtraction
- √(b² − 4ac) — the discriminant under the square root
Example: Solve 2x² − 5x + 3 = 0
a = 2, b = −5, c = 3
Discriminant = (−5)² − 4(2)(3) = 25 − 24 = 1 (positive, so two real roots)
x = (−(−5) ± √1) ÷ (2 × 2) = (5 ± 1) ÷ 4
x&sub1; = (5 + 1) ÷ 4 = 1.5 | x&sub2; = (5 − 1) ÷ 4 = 1.0
Verification: 2(1.5)² − 5(1.5) + 3 = 4.5 − 7.5 + 3 = 0 ✓
The calculator handles this automatically and shows the discriminant alongside the roots. The formula is shown here for transparency.
Where Quadratic Equations Appear
Projectile motion — The height of a thrown object over time follows a quadratic equation. If a ball is thrown upward with initial velocity of 20 m/s from ground level, its height h (in metres) at time t (in seconds) is h = −5t² + 20t. Setting h = 0 and solving gives the time the ball returns to the ground: t = 0 s (the start) and t = 4 s. Here a = −5, b = 20, c = 0.
Break-even analysis — A business's profit can sometimes be modelled as a quadratic function of units sold. If profit = −2x² + 40x − 150, setting this to zero and solving identifies the two sales volumes at which the business exactly breaks even. For more detailed break-even analysis, see our Break-Even Point Calculator.
Area and geometry problems — Many problems involving rectangular areas reduce to quadratic equations. If a rectangle has a perimeter of 40 cm and an area of 96 cm², expressing both constraints produces a quadratic that gives the side lengths.
Exam and competitive entrance preparation — Quadratic equations appear in CBSE Class 10 (Chapter 4), ICSE, and in JEE Main. The ability to quickly verify whether your roots are correct by checking the discriminant and substituting back is a useful exam technique.