How to read your quadratic results

A quadratic equation ax² + bx + c = 0 describes a parabola. Solving the equation means finding the x-values where that parabola crosses the x-axis. Those x-values are your roots (also called solutions or zeros).

• Δ = b² − 4ac tells you what kind of roots you have.
• Δ > 0: Two distinct real roots (the graph crosses the x-axis twice).
• Δ = 0: One real repeated root (the graph just touches the x-axis).
• Δ < 0: Two complex roots (the graph never crosses the x-axis).

• The axis of symmetry is x = −b / (2a).
• The vertex is the highest or lowest point of the parabola.
• If you’re graphing, the vertex is your anchor: everything mirrors around it.

Frequently Asked Questions

• What if a = 0?

  Then the equation becomes linear (bx + c = 0). This page is for quadratics only. If you enter a = 0, the solver will ask you to fix it.

• Why do I sometimes get complex roots?

  If the discriminant is negative, the square root of Δ involves i, the imaginary unit. That doesn’t mean you did anything wrong—your parabola simply doesn’t cross the x-axis.

• Can this give exact answers?

  Yes. When the discriminant is a perfect square (like 1, 4, 9, 16, …), the roots can often be written exactly using fractions and radicals. The solver shows both exact-style and decimal-style outputs.

• Is there a faster way than the quadratic formula?

  Sometimes. If the quadratic factors nicely (like x² − 3x + 2 = (x − 1)(x − 2)), factoring is quicker. Completing the square is also useful, especially for vertex form. The quadratic formula works for every quadratic, so it’s the universal tool.

• What does “double root” mean?

  When Δ = 0, both roots are the same number. The parabola touches the x-axis at exactly one point. That’s called a repeated (or double) root.

Quadratic formula, step-by-step (with examples)

A quadratic equation is any equation that can be written as ax² + bx + c = 0 where a ≠ 0. The “quadratic” part comes from the x² term. In the real world, quadratics show up everywhere: the arc of a basketball shot, the path of a water fountain, the way profit changes with price, or the geometry of a bridge arch. In school, quadratics show up because they’re the first time you need a systematic method to solve an equation that isn’t linear.

This page is designed to be a clean “one-stop solver”: you enter a, b, and c, and it gives you the roots (solutions), the discriminant, and helpful graph features like the vertex and axis of symmetry. But the real goal is understanding—so below we walk through the logic in a way that’s easy to remember and even easier to explain to someone else. When you can explain it, you own it.

The expression ax² + bx + c is a parabola when you graph it against x. Solving ax² + bx + c = 0 means finding the x-values where that parabola equals zero— i.e., where it crosses the x-axis. Those x-values are called roots or zeros. If the parabola crosses the axis twice, there are two real roots. If it just touches once, there’s one repeated root. If it never touches, the roots are complex (still valid solutions, just not x-axis crossings).

The discriminant is the expression Δ = b² − 4ac. It’s not magic—it’s what naturally appears when you derive the quadratic formula (more on that in a moment). Think of Δ as a “root personality test”:

• Δ > 0 → two distinct real solutions (two different x-intercepts).
• Δ = 0 → one real solution that repeats (the parabola is tangent to the x-axis).
• Δ < 0 → two complex conjugate solutions (no real x-intercepts).

That’s why this calculator highlights Δ. Once you know the sign of Δ, you already know the overall “shape” of the answer before computing any roots.

The quadratic formula is: x = (−b ± √(b² − 4ac)) / (2a). The ± symbol means you actually get two answers: one with + and one with − (unless Δ = 0, in which case both answers are the same).

If you ever forget the formula, remember its structure: “Negative b, plus or minus the square root, over 2a.” Everything else is inside the square root: b² − 4ac.

The formula isn’t a random thing someone invented. It comes from a standard technique called completing the square. Here’s the short version of the derivation:

1. Start with ax² + bx + c = 0.
2. Divide everything by a (because a ≠ 0): x² + (b/a)x + c/a = 0.
3. Move the constant: x² + (b/a)x = −c/a.
4. Add a number to both sides to make the left side a perfect square. The magic number is (b/2a)², because (x + b/2a)² = x² + (b/a)x + (b/2a)².
5. You now have: (x + b/2a)² = (b² − 4ac) / (4a²).
6. Take the square root of both sides, then solve for x. That produces the familiar (−b ± √(b² − 4ac)) / (2a).

Two takeaways: (1) the discriminant naturally appears during the algebra, and (2) the ± appears because square roots have a positive and negative version.

Example A: Two real roots — Solve x² − 3x + 2 = 0.

• a = 1, b = −3, c = 2
• Δ = b² − 4ac = (−3)² − 4(1)(2) = 9 − 8 = 1
• x = (−b ± √Δ)/(2a) = (3 ± 1)/2
• Roots: x₁ = (3 + 1)/2 = 2, and x₂ = (3 − 1)/2 = 1

Because Δ is a perfect square (1), the roots are clean integers. This is also a case where factoring is fast: (x − 1)(x − 2) = 0.

Example B: One repeated root — Solve x² + 6x + 9 = 0.

• a = 1, b = 6, c = 9
• Δ = 6² − 4(1)(9) = 36 − 36 = 0
• x = (−6 ± 0)/2 = −3

Only one x-value appears because the parabola just touches the x-axis. In fact, this quadratic is a perfect square: (x + 3)² = 0.

Example C: Complex roots — Solve 2x² + x + 5 = 0.

• a = 2, b = 1, c = 5
• Δ = 1² − 4(2)(5) = 1 − 40 = −39
• x = (−1 ± √(−39)) / 4 = (−1 ± i√39) / 4

The graph of 2x² + x + 5 never crosses the x-axis (because it’s always positive), but the complex solutions are still the correct algebraic solutions.

Beyond roots, quadratics have a useful geometric summary:

• Axis of symmetry: x = −b/(2a)
• Vertex: (h, k) where h = −b/(2a) and k = f(h)
• y-intercept: (0, c)

The vertex tells you the minimum (if a > 0) or maximum (if a < 0) value of the quadratic. This is why vertex form a(x − h)² + k is popular in graphing.

• Factoring is fastest when numbers are small and the quadratic “looks factorable.”
• Quadratic formula works for every case—use it when you want reliability.
• Completing the square is great when you want vertex form or to understand the shape.

• Write your a, b, c clearly before plugging into the formula.
• Put parentheses around b and around 2a. That prevents sign errors.
• Compute Δ as a separate line. It shows your reasoning and catches mistakes.
• If the answer must be exact, keep radicals (like √39) instead of decimals.

If you want a quick “sanity check,” plug each root back into ax² + bx + c. The result should be extremely close to zero (exactly zero for exact roots; very close for rounded decimals).

Quadratic toolkit (copy/paste)

• Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)
• Discriminant: Δ = b² − 4ac
• Axis of symmetry: x = −b / (2a)
• Vertex: (h, k) where h = −b/(2a), k = a h² + b h + c
• Vertex form: a(x − h)² + k
• Sum of roots: x₁ + x₂ = −b/a
• Product of roots: x₁ x₂ = c/a

Extra viral-friendly move: share the “toolkit” as a screenshot in your study group. It’s the whole chapter in one box.