Quadratic Formula Calculator
Solve ax² + bx + c = 0 instantly
How to Solve Quadratic Equations (Step by Step)
A quadratic equation is any equation that can be written in the standard form:
ax² + bx + c = 0 (where a ≠ 0)
The values of x that satisfy the equation are called the roots (or solutions). Our calculator above uses the famous quadratic formula to find those roots automatically, but understanding how it works is key for any algebra student.
The Quadratic Formula
The quadratic formula is derived by completing the square and gives the roots directly from the coefficients a, b, and c:
x = (-b ± √(b² - 4ac)) / (2a)
The expression under the square root, b² − 4ac, is called the discriminant (often represented by the Greek letter Δ). It decides the nature of the roots:
- Δ > 0 → Two distinct real roots.
- Δ = 0 → One real root (a double root).
- Δ < 0 → No real roots; instead there are two complex conjugate roots.
Example Walkthrough
Let’s solve 1x² − 5x + 6 = 0 manually, then you can test the calculator above with the same numbers.
- Identify coefficients: a = 1, b = -5, c = 6.
- Calculate the discriminant: Δ = (-5)² − 4×1×6 = 25 − 24 = 1.
- Since Δ > 0, there are two real roots.
- Apply the quadratic formula:
x = [5 ± √1] / (2×1) = [5 ± 1] / 2 - The roots are:
x₁ = (5 + 1)/2 = 3, x₂ = (5 − 1)/2 = 2
Try it in the calculator: enter a = 1, b = -5, c = 6 and click “Calculate Roots”. You’ll see the discriminant, steps, and the same answers.
When Do You Use the Quadratic Formula?
The quadratic formula works for every quadratic equation, even when factoring is difficult or impossible (when roots are irrational or complex). It’s especially useful when:
- Factoring doesn’t come easily (e.g., 2x² − 4x − 3 = 0).
- The coefficients are decimals or large numbers.
- You need exact answers (including square roots) rather than decimal approximations.
Why Our Calculator Helps
This tool doesn’t just spit out the answers—it shows the discriminant, the step‑by‑step substitution into the quadratic formula, and clearly labels whether the roots are real or complex. Use it to check your homework, learn the pattern, or quickly solve any quadratic equation on the go.
💡 Pro tip: Always double‑check that a is not zero. If a = 0, the equation is linear, not quadratic, and the quadratic formula doesn’t apply (our calculator will warn you about that).
Feel free to bookmark this page – it’s completely free and works instantly right inside your browser. Happy solving!
