Quadratic Formula

The formula

For any quadratic equation ax² + bx + c = 0 with a ≠ 0, the solutions are:

x = (−b ± √(b² − 4ac)) / 2a

The ± gives two solutions — one with +, one with −. The expression under the square root, b² − 4ac, is called the discriminant and tells you what kind of solutions to expect before computing them.

The discriminant — three cases

Discriminant Δ = b² − 4ac	Number of real roots	Geometric meaning
Δ > 0	Two distinct real roots	Parabola crosses the x-axis at two points
Δ = 0	One real root (repeated)	Parabola touches the x-axis at exactly one point (vertex)
Δ < 0	Zero real roots; two complex conjugate roots	Parabola doesn't touch the x-axis

The complex roots in the Δ < 0 case have the form (−b ± i√|Δ|) / 2a. They come in conjugate pairs because the original coefficients are real.

Derivation by completing the square

Start with the general quadratic:

ax² + bx + c = 0

Step 1 — divide by a (we can because a ≠ 0):

x² + (b/a)x + c/a = 0

Step 2 — move c/a to the right:

x² + (b/a)x = −c/a

Step 3 — complete the square. Add (b/2a)² to both sides:

x² + (b/a)x + (b/2a)² = (b/2a)² − c/a

Step 4 — factor the left side as a perfect square:

(x + b/2a)² = (b² − 4ac) / (4a²)

Step 5 — take square roots and solve:

x + b/2a = ± √(b² − 4ac) / (2a)
x = (−b ± √(b² − 4ac)) / (2a)

That's the formula, derived in five lines. The square root in step 5 is what produces the ±, and what creates the discriminant inside.

Worked examples

Example 1 — two real roots

Solve x² − 5x + 6 = 0. Here a = 1, b = −5, c = 6.

Δ = (−5)² − 4·1·6 = 25 − 24 = 1
x = (5 ± √1) / 2 = (5 ± 1) / 2
x = 3 or x = 2

Verify by factoring — x² − 5x + 6 = (x − 2)(x − 3). ✓

Example 2 — repeated root

Solve 4x² − 12x + 9 = 0. Here a = 4, b = −12, c = 9.

Δ = (−12)² − 4·4·9 = 144 − 144 = 0
x = 12 / (2·4) = 3/2

One repeated root. Verify — 4x² − 12x + 9 = (2x − 3)².

Example 3 — complex roots

Solve x² + x + 1 = 0. Here a = 1, b = 1, c = 1.

Δ = 1 − 4 = −3
x = (−1 ± √(−3)) / 2 = (−1 ± i√3) / 2

Two complex conjugate roots. These are the non-real cube roots of 1 (besides 1 itself).

Vieta's formulas — sum and product of roots

If r₁ and r₂ are the roots of ax² + bx + c = 0, then:

r₁ + r₂ = −b/a
r₁ · r₂ = c/a

You can verify by adding/multiplying the formula's two output values. These let you construct a quadratic with desired roots — if you want roots 3 and 5, take a = 1, then b = −(3+5) = −8 and c = 3×5 = 15. The quadratic is x² − 8x + 15.

JavaScript: numerically stable solver

function quadratic(a, b, c) {
  if (a === 0) {
    if (b === 0) return c === 0 ? 'infinite solutions' : 'no solution';
    return [-c / b];  // linear case
  }
  const disc = b * b - 4 * a * c;
  if (disc > 0) {
    // Stable formula — avoid catastrophic cancellation
    const sqrtD = Math.sqrt(disc);
    // Pick the larger-magnitude term in numerator
    const q = -0.5 * (b + Math.sign(b) * sqrtD);
    return [q / a, c / q];
  } else if (disc === 0) {
    return [-b / (2 * a)];
  } else {
    // Complex roots
    const real = -b / (2 * a);
    const imag = Math.sqrt(-disc) / (2 * a);
    return [{ re: real, im: imag }, { re: real, im: -imag }];
  }
}

quadratic(1, -5, 6);   // [3, 2]
quadratic(4, -12, 9);  // [1.5]
quadratic(1, 1, 1);    // [{re: -0.5, im: 0.866...}, {re: -0.5, im: -0.866...}]

The "stable" form handles cases like a = 1, b = -10⁸, c = 1, where the naive formula loses ~7 digits of precision. Vieta's relation r₁ × r₂ = c/a lets us compute the smaller root by division instead of subtraction.

Python implementation

import cmath

def quadratic(a, b, c):
    """Returns roots of ax² + bx + c = 0 as a tuple."""
    if a == 0:
        return () if b == 0 else (-c / b,)
    disc = b*b - 4*a*c
    sqrtD = cmath.sqrt(disc)
    return ((-b + sqrtD) / (2*a), (-b - sqrtD) / (2*a))

quadratic(1, -5, 6)   # ((3+0j), (2+0j))
quadratic(1, 1, 1)    # ((-0.5+0.866j), (-0.5-0.866j))

Python's cmath module handles complex arithmetic natively. cmath.sqrt(-3) returns 1.732j instead of throwing.

Quadratic formula vs other methods

Method	When to use	Advantages	Drawbacks
Quadratic formula	Always (when you need exact roots)	Always works; gives both roots; reveals discriminant	Tedious for "easy" cases
Factoring	Integer or simple rational roots	Fast; reveals structure	Can't factor when roots aren't rational
Completing the square	Want vertex form for graphing	Gives vertex (h, k) directly	More steps than the formula
Graphical / numerical	Don't need exact answer	Visual intuition; works for any function	Approximate

When the formula doesn't apply

• a = 0. The equation isn't quadratic; it's linear. The formula's denominator is zero. Solve as bx + c = 0.
• Higher-degree polynomials. ax³ + bx² + cx + d = 0 needs the cubic formula or numerical methods. The "quadratic formula" is specifically for degree 2.
• Non-polynomial equations. e^x = sin(x), x² + 2^x = 5 — transcendental equations have no algebraic formula. Use numerical root-finding (Newton's method, bisection).
• Equations in multiple variables. x² + y² = 1 (a circle, not a single quadratic). The formula solves for one variable at a time, treating others as parameters.

Common mistakes

• Sign errors on b. The formula has −b, not +b. x² − 5x + 6 = 0 has b = −5, so −b = +5 — easy to drop a sign.
• Forgetting to divide everything by 2a. The denominator applies to the whole numerator, not just one term. Brackets matter.
• Computing the discriminant wrong. b² − 4ac, not (b−4ac)² or b·b − 4·a·c (those are the same; the issue is operator precedence in pen-and-paper).
• Square-rooting a negative without going to complex. When discriminant is negative, the answers are complex. Some calculators just say "ERROR"; you need to switch to complex mode or use cmath.
• Catastrophic cancellation in floating-point. When b² is much larger than 4ac, the naive formula loses precision. Use the numerically stable form (Vieta's relation) for production code.
• Confusing roots with x-intercepts. If a quadratic has complex roots (Δ < 0), there are no x-intercepts on the real plane — the parabola never touches the x-axis. The complex roots aren't visible on a real-number graph.