How to Solve Any Quadratic Equation: The Quadratic Formula, Factoring, and Completing the Square

A quadratic equation — anything of the form ax² + bx + c = 0 — shows up constantly in physics, engineering, and finance: projectile paths, profit maximization, break-even points. There are three standard ways to solve one, and knowing which to pick saves you time on exams and in real calculations. The universal fallback is the quadratic formula, which works on every quadratic, no matter how messy.

In a hurry? Enter your coefficients in the Algebra Calculator and you get the roots, the discriminant, and a worked solution instantly.

The Quadratic Formula (Always Works)

For ax² + bx + c = 0, the roots are:

x = (−b ± √(b² − 4ac)) / 2a

The part under the square root — b² − 4ac — is the discriminant, and it tells you what kind of solutions to expect before you even finish the calculation:

Discriminant (D = b² − 4ac)	Number of Real Roots	What It Means
D > 0	2	Two distinct real roots (parabola crosses the x-axis twice)
D = 0	1	One repeated root (parabola touches the x-axis once)
D < 0	0 real	Two complex roots (parabola never touches the x-axis)

Worked Example: 2x² + 4x − 6 = 0

1. Identify a, b, c: a = 2, b = 4, c = −6
2. Compute the discriminant: D = 4² − 4(2)(−6) = 16 + 48 = 64
3. Apply the formula: x = (−4 ± √64) / 4 = (−4 ± 8) / 4
4. Simplify: x = 1 and x = −3

Check: plug x = 1 back in — 2(1) + 4(1) − 6 = 0. Correct.

When Factoring Is Faster

If a, b, and c are small integers and the equation factors cleanly, factoring is quicker. Take x² − 5x + 6 = 0: you need two numbers that multiply to 6 and add to −5, which are −2 and −3. So (x − 2)(x − 3) = 0, giving x = 2 and x = 3. Use factoring when the leading coefficient is 1 (or a small number) and you can spot the pair in a few seconds. When in doubt, the quadratic formula never lies.

When Completing the Square Helps

Completing the square rewrites ax² + bx + c = 0 as a(x − h)² + k = 0, which reveals the parabola’s vertex (h, k) directly. It is the method of choice when you need the maximum or minimum value — like finding the peak height of a projectile or the price that maximizes revenue. It is more algebra than the formula, but it hands you the vertex for free.

Real-World Uses

• Projectile motion: h(t) = −16t² + vt + h₀ describes how high a ball gets and when it lands
• Profit optimization: Revenue and cost curves are quadratics; the difference peaks at the vertex
• Break-even analysis: Setting profit = 0 gives a quadratic whose roots are your break-even points
• Geometry: Area problems like maximizing a rectangular enclosure with fixed fencing

For any quadratic, the fastest path to a verified answer is the Algebra Calculator — enter a, b, and c and it returns the roots, discriminant, and step-by-step reasoning. Use it to check your manual work until the formula feels automatic.