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Math & School
Quadratic Formula Calculator — Solve ax² + bx + c = 0 with Steps
Solve any quadratic equation and see the discriminant, both roots and the vertex
The quadratic formula is the one tool that solves every quadratic equation — no factoring, no guessing, no completing the square by hand. Any equation that can be written as ax² + bx + c = 0 (with a ≠ 0) is solved by:
x = ( −b ± √(b² − 4ac) ) / (2a)
That single line is the backbone of Algebra 1, Algebra 2, precalculus and the SAT/ACT math sections, and it shows up far beyond the classroom — projectile motion, area and optimization problems, the path of a thrown ball, break-even analysis and the shape of every parabola.
The term under the square root, b² − 4ac, is called the discriminant (often written D or Δ), and it tells you the nature of the solutions before you finish solving:
- D > 0 → two distinct real roots (the parabola crosses the x-axis twice).
- D = 0 → one repeated real root (the parabola just touches the x-axis at its vertex).
- D < 0 → two complex-conjugate roots of the form p ± qi (the parabola never touches the x-axis).
Here's a clean worked example. Solve x² − 3x + 2 = 0, so a = 1, b = −3, c = 2. The discriminant is D = (−3)² − 4(1)(2) = 9 − 8 = 1, which is positive, so expect two real roots. Plugging in: x = (3 ± √1) / 2 = (3 ± 1) / 2, giving x = 2 and x = 1. You can check: 2 + 1 = 3 = −b/a and 2 × 1 = 2 = c/a, which is Vieta's formula and a fast way to verify any quadratic.
This calculator does all of it at once. Type your a, b and c and it returns both roots (real or complex), the value of the discriminant with its meaning, the vertex (−b/2a, the minimum or maximum point), the axis of symmetry, and a step-by-step substitution so you can follow the work or check your homework. It also handles the edge cases cleanly: if a = 0 the equation is actually linear, and it solves that for you instead of dividing by zero.
Whether you're checking a factoring answer, solving an equation that won't factor nicely, or finding where a parabola crosses the axis, the quadratic formula always works — and this tool shows every step.
📖 See also: The Quadratic Formula, Explained Step by Step
Calculator
Fill in the fields and click "Calculate" for instant results.
📰 Formula
• Quadratic formula: x = ( −b ± √(b² − 4ac) ) / (2a) • Discriminant: D = b² − 4ac • D > 0 → two real roots · D = 0 → one repeated root · D < 0 → complex roots p ± qi • Complex roots: p = −b / (2a), q = √(−D) / (2a) • Vertex: x = −b / (2a), y = c − b² / (4a) • Vieta's check: sum of roots = −b/a, product of roots = c/a
📰 Formula
• Quadratic formula: x = ( −b ± √(b² − 4ac) ) / (2a) • Discriminant: D = b² − 4ac • D > 0 → two real roots · D = 0 → one repeated root · D < 0 → complex roots p ± qi • Complex roots: p = −b / (2a), q = √(−D) / (2a) • Vertex: x = −b / (2a), y = c − b² / (4a) • Vieta's check: sum of roots = −b/a, product of roots = c/a
🧪 Worked examples
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Example 2
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Example 3
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Example 4
⚠️ Common mistakes
- Dropping the negative sign on −b, or forgetting to square b before subtracting 4ac.
- Computing 4ac without the correct signs (a negative c flips the sign inside the radical).
- Dividing only the radical by 2a instead of dividing the whole numerator (−b ± √D) by 2a.
- Calling the equation 'no solution' when D < 0 — there are still two complex roots.
- Forgetting that a ≠ 0; if a = 0 the equation is linear, not quadratic.
💡 Tips
- Compute the discriminant b² − 4ac first — it tells you what kind of answer to expect before you finish.
- Use Vieta's formulas to check: the two roots should add to −b/a and multiply to c/a.
- If a, b and c share a common factor, dividing it out first keeps the numbers small (it doesn't change the roots).
- The vertex x-coordinate is exactly halfway between the two real roots, at x = −b/(2a).
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❓ Frequently asked questions
What is the quadratic formula?
It's x = (−b ± √(b² − 4ac)) / (2a), and it solves any equation written as ax² + bx + c = 0 where a is not zero. The ± gives the two solutions, one using plus and one using minus.
How do I use this quadratic formula calculator?
Enter the three coefficients a, b and c from your equation ax² + bx + c = 0. The calculator returns both roots, the discriminant, the vertex and a step-by-step solution. Make sure to include the correct signs — for example, x² − 3x + 2 has b = −3 and c = +2.
What is the discriminant and why does it matter?
The discriminant is b² − 4ac, the part under the square root. Its sign tells you the type of solutions: positive means two real roots, zero means one repeated real root, and negative means two complex roots. You can read the nature of the answer before solving.
What does it mean when the discriminant is negative?
A negative discriminant means there are no real solutions, but there are two complex-conjugate roots of the form p ± qi. The parabola does not cross the x-axis. This calculator reports those complex roots rather than saying 'no solution.'
Can the quadratic formula solve equations I can't factor?
Yes — that's its main strength. Factoring only works for 'nice' equations, but the quadratic formula solves every quadratic, including ones with irrational or complex roots. It always gives the exact answer.
What if a = 0?
If a = 0 the equation is not quadratic — it's linear (bx + c = 0), and dividing by 2a would mean dividing by zero. This calculator detects that case and solves the linear equation x = −c/b instead.
What is the vertex of a quadratic and how do I find it?
The vertex is the highest or lowest point of the parabola. Its x-coordinate is −b/(2a), and you get the y-coordinate by plugging that x back into the equation (y = c − b²/4a). The vertex sits exactly midway between the two real roots.
How can I check my quadratic answer is correct?
Use Vieta's formulas: the two roots should add up to −b/a and multiply to c/a. For x² − 3x + 2 = 0 the roots 2 and 1 add to 3 (= −b/a) and multiply to 2 (= c/a), confirming the solution.
Is the quadratic formula the same as completing the square?
The quadratic formula is derived by completing the square on the general equation ax² + bx + c = 0, so they give the same answers. The formula is just the finished, ready-to-use result — you plug in a, b and c instead of doing the algebra each time.