Free Quadratic Formula Calculator Online
This free online quadratic formula calculator solves any quadratic equation ax² + bx + c = 0. Enter the three coefficients to get both roots — real or complex — with the discriminant shown.
Equation: ax² + bx + c = 0
Enter the three coefficients below.
Quadratic Roots
Enter coefficients and click Solve
The Quadratic Formula Explained
The quadratic formula solves any quadratic equation of the form ax² + bx + c = 0. This free quadratic formula calculator gives you both roots directly — no factoring or guessing needed. Calculate online and see the full solution instantly:
x = (−b ± √(b² − 4ac)) / (2a)
The ± means there are generally two solutions: one with +√ and one with −√. The quadratic formula calculator computes both.
The expression inside the square root — b² − 4ac — is called the discriminant. It determines the nature of the roots before you even complete the calculation. A positive discriminant means two different real roots (the most common case). Zero means exactly one root (the parabola touches the x-axis at exactly one point). Negative means two complex roots (the parabola doesn't cross the x-axis at all).
The quadratic formula works for every quadratic equation — even ones that can't be factored with integers. Equations like x² + 3x − 7 = 0 don't factor neatly, but the quadratic formula calculator handles them in one step.
Discriminant Reference
| Discriminant | Root Type | Example |
|---|---|---|
| b² − 4ac > 0 | Two distinct real roots | x² − 5x + 6 = 0 → x = 2, x = 3 |
| b² − 4ac = 0 | One repeated real root | x² − 4x + 4 = 0 → x = 2 (double) |
| b² − 4ac < 0 | Two complex conjugate roots | x² + x + 1 = 0 → x = −½ ± (√3/2)i |
Worked Example: x² − 5x + 6 = 0
a = 1, b = −5, c = 6
Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1
x = (5 ± √1) / 2 = (5 ± 1) / 2
x₁ = 3, x₂ = 2
Where the Quadratic Formula Appears
Projectile Motion
The height of a thrown ball follows h = −16t² + v₀t + h₀ (in feet). To find when it hits the ground, set h=0 and use the quadratic formula. This is exactly the form our free online quadratic formula calculator handles — enter the coefficients and get the time.
Area Problems
A rectangular field is 3 meters longer than it is wide. Its area is 180 m². Setting up the equation: w(w+3) = 180, which rearranges to w² + 3w − 180 = 0. Enter a=1, b=3, c=−180 into this free quadratic formula calculator online for the exact width.
Break-Even Analysis
Revenue and cost curves that involve quadratic terms require the quadratic formula to find the break-even points. A revenue model like R = −2p² + 120p and cost C = 400 gives −2p² + 120p − 400 = 0 — directly solvable with this free online quadratic formula calculator.
Quadratic Formula Tips
Put It in Standard Form First
The quadratic formula requires ax² + bx + c = 0. Move all terms to one side first. For 3x² = 2x − 5: rewrite as 3x² − 2x + 5 = 0, giving a=3, b=−2, c=5.
Check the Discriminant First
Calculate b²−4ac before the full quadratic formula. This tells you immediately whether to expect real roots or complex ones — saving you from surprise when the quadratic formula calculator shows imaginary numbers.
Verify by Substitution
After using the quadratic formula, plug each root back in. For x² − 5x + 6 = 0, substituting x=3 gives 9−15+6 = 0. Substituting x=2 gives 4−10+6 = 0. Both check out.
Factoring vs. Quadratic Formula
Factoring is faster when integer roots are obvious. The quadratic formula works for everything — including equations with irrational or complex roots that won't factor with integers. When in doubt, use this free quadratic formula calculator online.
Frequently Asked Questions
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