Quadratic Equation Solver
Solve ax²+bx+c=0 with steps
Working out solve ax²+bx+c=0 with steps is easier with Quadratic Equation Solver — a free tool that does the math for you. Fill in a, b and c and read your answer immediately. The calculation is displayed with all its working, so the number always makes sense. A practical tool for students, professionals, and everyday planners alike. Everything runs in your browser — your inputs are not sent to our servers, and it works offline after the first visit (currency conversion needs a live connection). It is part of the Math collection on CalcProMaster, alongside quadratic equation solver with steps, solve quadratic equations by factoring calculator and more. Great for comparing scenarios — change a value and watch the impact immediately. Bookmark it and the answer is always one click away.
What does the Quadratic Equation Solver do?
Quadratic Equation Solver works out the ax²+bx+c=0 from the a, b, and c, following standard Math conventions — the page defaults produce a ax²+bx+c=0 of x₁=3.0000, x₂=2.0000.
- Inputs: a, b, and c.
- Output: the ax²+bx+c=0, plus the intermediate steps behind it.
- Method: the standard Math formula, evaluated entirely in your browser.
Quick answer
With the default inputs (a of 1, b of -5, c of 6), quadratic equation solver returns a ax²+bx+c=0 of x₁=3.0000, x₂=2.0000. Assumptions and limits are summarized below.
How does it work?
Quadratic Equation Solver computes the ax²+bx+c=0 directly from your inputs — the a, b, and c feed the formula. Nothing is uploaded: the math runs locally in your browser and the result appears as you type.
What This Calculator Really Does
Every quadratic equation — anything of the form ax² + bx + c = 0 — has exactly two solutions once you count complex numbers. This solver applies the quadratic formula, shows the discriminant first (because it predicts what kind of roots you will get), and displays both roots with the arithmetic visible.
The discriminant is the strategic shortcut: b² − 4ac tells you before any division whether you will meet two real roots, one repeated root, or a conjugate pair of complex ones. Reading it first turns the formula from symbol-pushing into understanding.
All computation is local to your browser — nothing uploaded, nothing tracked, offline after the first visit.
Quick answer: 2x² − 7x + 3 = 0 has discriminant 25, so x = (7 ± 5) ÷ 4 → x = 3 and x = 0.5. Substitute either back: the left side becomes 0.
The Formula, Explained Plainly
x = (−b ± √(b² − 4ac)) ÷ 2a
- a, b, c — the coefficients of ax² + bx + c = 0; a must be nonzero or the equation is not quadratic.
- Discriminant D = b² − 4ac — D > 0: two distinct real roots; D = 0: one repeated real root; D < 0: two complex conjugate roots.
Useful identities: the two roots always sum to −b/a and multiply to c/a — quick checks that catch sign slips before they propagate.
How to Use It — In Order
- Rearrange to standard form — everything on one side, = 0 on the other.
- Identify a, b and c — keeping their signs; −7 is not 7.
- Enter the coefficients — a cannot be zero.
- Read the discriminant — it forecasts two real, one repeated, or complex roots.
- Read both roots — with the formula steps shown.
- Verify by substitution — plug each root into the original equation; both should give 0.
Worked Example, Verified by Hand
Solve 2x² − 7x + 3 = 0 (a = 2, b = −7, c = 3):
- Discriminant: b² − 4ac = 49 − 24 = 25 — positive, so two real roots.
- √25 = 5.
- x = (7 ± 5) ÷ 4 → x = 12 ÷ 4 = 3, and x = 2 ÷ 4 = 0.5.
- Check x = 3: 2(9) − 7(3) + 3 = 18 − 21 + 3 = 0 ✓. Check x = 0.5: 2(0.25) − 3.5 + 3 = 0 ✓.
Sum check: roots 3 + 0.5 = 3.5 = −b/a = 7/2 ✓. Product: 1.5 = c/a = 3/2 ✓. Three independent confirmations.
Common Mistakes to Avoid
- Sign slips on b — if the equation shows −7x, then b = −7 and −b = +7 inside the formula.
- Forgetting to move everything to one side first — 2x² = 7x − 3 is not in standard form yet.
- Dividing only the square root by 2a — the entire numerator, −b ± √D, is divided by 2a.
- Accepting a = 0 — then the equation is linear and the quadratic formula does not apply.
- Declaring "no solution" when D < 0 — complex roots exist; they are just not real numbers.
Limitations
- Quadratics only — cubics and higher need different machinery (the cubic-equation solver covers those).
- Floating-point cancellation can lose precision when b² vastly exceeds 4ac; the identities above help verify.
- Rounding of irrational roots is display-only; the calculator keeps full precision internally.
Expected Accuracy
Roots are computed at full floating-point precision and rounded for display. Rational roots (as in the example) are exact; irrational roots match hand computation to displayed precision.
Privacy — Your Data Never Leaves This Device
Everything is computed locally: no inputs stored or sent, no tracking — the solver works offline too.
Related Tools & Guides
Natural next steps from Quadratic:
- All Math Calculators
- Cubic Equation Solver — third-degree polynomials
- Completing the Square — the derivation behind the formula
- System of Linear Equations — when degree is one
- Logarithm Calculator — another inverse-operation solver
Sources & Standards
The quadratic formula is standard secondary-school mathematics; derivation via completing the square is universal.
- Wolfram MathWorld — Quadratic Equation (reference)
Bottom Line
Discriminant first, formula second, substitution check last — three steps that make every quadratic routine. The sum and product identities are your built-in error alarms.
From Our Guides Library
Frequently Asked Questions
What is the quadratic formula?
For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a. It gives both roots of any quadratic with a ≠ 0.
How do I solve 2x² − 7x + 3 = 0?
Discriminant = 49 − 24 = 25, √25 = 5, so x = (7 ± 5)/4, giving x = 3 and x = 0.5. Both check by substitution.
What does the discriminant tell me?
b² − 4ac: positive means two distinct real roots, zero means one repeated root, negative means two complex conjugate roots — before you compute anything else.
What if the discriminant is negative?
The roots are complex conjugates: (−b ± i√|D|) ÷ 2a. There are no real solutions, but the equation still has exactly two roots over the complex numbers.
Can a quadratic have only one root?
Yes — when the discriminant is exactly zero the two roots coincide: x = −b ÷ 2a, called a double or repeated root.
How do I check my quadratic roots quickly?
Their sum should equal −b/a and their product c/a. For the example: 3 + 0.5 = 3.5 = 7/2 and 3 × 0.5 = 1.5 = 3/2.
Is this solver free and private?
Yes — it runs entirely in your browser with no sign-up, no upload and no tracking.
What does the Quadratic Equation Solver calculate?
At its core, Quadratic Equation Solver takes the a, b, and c and evaluates the standard formula step by step, so the output can be checked rather than trusted on faith. Because the working is visible: Quadratic Equation Solver shows each operation behind the ax²+bx+c=0 in the steps panel, so you can verify the result instead of trusting a black box.
How is the ax²+bx+c=0 calculated?
The first steps are equation: 1x² + -5x + 6 = 0, then discriminant d = b² - 4ac = -5² - 4×1×6 = 1. Quadratic Equation Solver lists every intermediate step in the result panel, so the derivation of the ax²+bx+c=0 can be checked line by line.
What do I need to use the Quadratic Equation Solver?
The a, b, and c it asks for, or the page defaults if you just want to see the calculation work. Each input maps directly to the formula, and changing any one of them recalculates the ax²+bx+c=0 instantly.
What does the result from the Quadratic Equation Solver mean?
The main number the quadratic equation solver returns is the ax²+bx+c=0 for your exact inputs, and the supporting figures and step list give it context. The model behind Quadratic Equation Solver covers the standard case; special cases, edge values, or jurisdiction-specific rules may need manual adjustment.
When is the Quadratic Equation Solver most useful?
Quadratic Equation Solver fits planning and checking: planning around a target figure, comparing scenarios side by side, and double-checking the ax²+bx+c=0, or any moment when the result needs to be right the first time. Run Quadratic Equation Solver twice with deliberately low and high inputs; the spread tells you how sensitive the figure is, which a single run never shows.