The quadratic formula
Any quadratic equation ax² + bx + c = 0 (with a ≠ 0) can be solved in one step using the quadratic formula.
The quadratic formula is: x = (−b ± √(b² − 4ac)) / (2a). The ± gives you the two roots. You plug in a, b, c and the formula returns both solutions directly.
Two things make it "one step": it works for every quadratic (factoring only works when the roots are nice), and it handles complex roots automatically when the discriminant is negative.
Derivation (for interest, not memorization): the formula comes from completing the square on ax² + bx + c = 0. The calculator does the arithmetic — your job is to identify a, b, c correctly from the equation.
- a = 1, b = −5, c = 6
- Discriminant: (−5)² − 4·1·6 = 25 − 24 = 1
- √1 = 1
- x = (5 ± 1) / 2
- x = 3 or x = 2
The formula is one line. Identify a, b, c, compute the discriminant, then apply ± √Δ.
What the discriminant (Δ = b² − 4ac) tells you
The sign of the discriminant classifies the roots before you compute them.
Δ > 0: two distinct real roots. The parabola crosses the x-axis at two points.
Δ = 0: one repeated real root (double root). The parabola touches the x-axis at exactly one point — the vertex.
Δ < 0: two complex conjugate roots of the form p ± qi. The parabola sits entirely above or below the x-axis and never crosses it.
In a physics or engineering problem, negative Δ usually means "no real solution" — e.g., a projectile never reaches a given height, or a circuit never oscillates at the requested frequency.
- x² − 3x + 2: Δ = 1 (>0) → real roots 1, 2
- x² − 4x + 4: Δ = 0 → double root x = 2
- x² + x + 1: Δ = −3 (<0) → complex roots −½ ± (√3/2)i
Check Δ first — it tells you what kind of roots to expect before you compute them.
Vertex & the shape of the parabola
The vertex of y = ax² + bx + c sits at x = −b/(2a). Its y-value and the sign of a fully describe the parabola.
Vertex x-coordinate: always at −b/(2a). This is also the axis of symmetry — the parabola is a mirror image on either side.
Vertex y-coordinate: plug the vertex x back into f(x) = ax² + bx + c. That gives the minimum (if a > 0) or maximum (if a < 0) of the parabola.
Sign of a: positive a means the parabola opens upward (a smile shape). Negative a means it opens downward (a frown shape). The calculator marks the vertex and roots directly on the plot so you can see this instantly.
Real-world: the vertex is where motion reverses (a ball thrown up starts falling), where profit peaks before declining, or where a cable sags the most.
Vertex at x = −b/(2a). Sign of a → opens up (+) or down (−). Use the calculator plot to see it.
When does the factored form work?
The factored form (x − r₁)(x − r₂) only reads cleanly when the roots are rational numbers. For irrational or complex roots, the formula is your only route.
If Δ is a perfect square of a rational number (e.g., 1, 4, 9, 25, or 1/4), both roots are rational and the equation factors neatly over the integers or simple fractions.
If Δ is positive but not a perfect square (e.g., 5, 7, 12), the roots are irrational. You get answers like x = (3 ± √5) / 2 — still correct, just not factorable over the rationals.
If Δ is negative, the roots are complex. No real factored form exists; you need complex numbers to write (x − (p + qi))(x − (p − qi)).
- x² − 5x + 6 = 0 → Δ = 1 (perfect square) → (x − 2)(x − 3) ✓
- x² − 3x + 1 = 0 → Δ = 5 → x = (3 ± √5) / 2, no rational factor form
- x² + 1 = 0 → Δ = −4 → x = ±i, complex only
The calculator shows the factored form only when it fits cleanly. If it's omitted, the roots are irrational or complex.
Five sign-error traps to avoid
Most quadratic mistakes are sign errors in the discriminant. Work carefully with negative coefficients.
1. Losing a parenthesis on negative b. If b = −5, then b² = (−5)² = 25, not −25. Always parenthesize b when squaring.
2. Confusing −b with b. The formula starts with −b, so if b = −5, then −b = +5. Students often substitute b directly, forgetting the sign flip.
3. Sign error inside 4ac. If c is negative, 4ac is negative, and −4ac becomes positive. Double-check the sign of the final Δ.
4. Forgetting the ± gives two values. Always write out both roots: one with +, one with −. The answer isn't a single number.
5. Applying the formula to non-quadratics. If a = 0, the equation is linear — the formula divides by zero and breaks. The solver catches this and errors out.
Parenthesize b, watch the sign of −b, double-check the 4ac sign, list both ± roots, and confirm a ≠ 0.
Frequently Asked Questions
Does this solver handle complex roots?
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Does this solver handle complex roots?
▾Yes. When the discriminant is negative, the calculator returns the complex conjugate pair in the form p ± qi. The parabola plot still renders (it just sits entirely above or below the x-axis, with no root markers).
Why is there no factored form for some equations?
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Why is there no factored form for some equations?
▾The factored form (x − r₁)(x − r₂) only fits cleanly when the roots are rational numbers. If your equation has irrational roots (e.g., involving √5) or complex roots, the factored form would require irrational or complex factors — the calculator omits it in those cases rather than showing a messy expression.
What is the vertex used for?
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What is the vertex used for?
▾The vertex is the parabola's minimum (if it opens upward) or maximum (if it opens downward). In optimization problems — maximum profit, peak projectile height, lowest cost — the vertex is the answer you're looking for, not the roots.
Can I enter fractional coefficients?
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Can I enter fractional coefficients?
▾Yes. The inputs accept decimals (e.g., a = 0.5, c = 2.75). Integer inputs produce cleaner factored forms and vertex coordinates; decimals work but may produce slightly irrational-looking intermediate numbers.
Why does the solver reject a = 0?
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Why does the solver reject a = 0?
▾With a = 0, the equation simplifies to bx + c = 0 — a linear equation, not quadratic. The quadratic formula divides by 2a, so a = 0 would be a division by zero. Use a linear equation solver (or just compute x = −c/b) for that case.
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Enter a, b, c — see the discriminant, roots, vertex, factored form, and a live parabola plot with every step shown.
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