A quadratic equation is just an equation where the highest power is 2 — but that single squared term creates a curve, two possible answers, and a toolkit of solving methods. Here's how they all fit together.
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are numbers and a is not zero. The graph of a quadratic is a U-shaped curve called a parabola, and solving the equation means finding the x-values where that parabola crosses the x-axis — the roots.
Standard form
ax² + bx + c = 0, with a ≠ 0. Everything else is built from this shape.
Parabola
Opens up if a > 0, down if a < 0. The roots are where it crosses y = 0.
Three main methods
Factoring, completing the square, and the quadratic formula.
The Standard Form and What Each Part Does
ANATOMY OF A QUADRATICEvery quadratic equation can be rearranged into standard form. The coefficients aren't just decoration — each one controls a different part of the curve.
- If a > 0, the parabola opens upward and has a minimum point (the vertex).
- If a < 0, the parabola opens downward and has a maximum point.
- If a = 0, it's not quadratic at all — it becomes a straight line.
- The vertex sits at x = −b / 2a, and the axis of symmetry is the vertical line through it.
Before you solve anything, checking the sign of a tells you whether the parabola opens up or down — and therefore whether the quadratic has a minimum or a maximum. That's often the whole answer in real-world optimization problems.
Method 1 — Factoring
FASTEST WHEN IT WORKSFactoring rewrites the quadratic as a product of two simpler expressions. If (x − r)(x − s) = 0, then either x = r or x = s — because a product is zero only when one of its factors is zero.
Step-by-step example: x² + 5x + 6 = 0
- Look for two numbers that multiply to a·c = 1 × 6 = 6 and add to b = 5. Those numbers are 2 and 3.
- Write the factored form: (x + 2)(x + 3) = 0.
- Set each factor to zero: x + 2 = 0 or x + 3 = 0.
- Solve: x = −2 or x = −3. These are the two roots.
Factoring is quick when the numbers are friendly, but it doesn't always work with integers. When it doesn't, move to one of the universal methods below.
| Quadratic | Factored form | Roots |
|---|---|---|
| x² + 5x + 6 | (x + 2)(x + 3) | x = −2, −3 |
| x² − 9 | (x − 3)(x + 3) | x = 3, −3 |
| x² − 4x + 4 | (x − 2)² | x = 2 (double root) |
| 2x² + 7x + 3 | (2x + 1)(x + 3) | x = −1/2, −3 |
Method 2 — Completing the Square
TURNS ANY QUADRATIC INTO VERTEX FORMCompleting the square rewrites ax² + bx + c as a perfect square plus a constant. It's the method behind the quadratic formula, and it's the cleanest way to find the vertex of a parabola.
Step-by-step example: x² + 6x + 2 = 0
- Move the constant to the other side: x² + 6x = −2.
- Take half of b (which is 3) and square it (9). Add that to both sides: x² + 6x + 9 = 7.
- Write the left side as a perfect square: (x + 3)² = 7.
- Take the square root of both sides: x + 3 = ±√7.
- Solve: x = −3 ± √7.
Completing the square always works, even when the roots are irrational. It's also the bridge to the vertex form y = a(x − h)² + k, where (h, k) is the vertex.
Use completing the square when you need the vertex, when the equation has a perfect-square pattern, or when you want to understand where the quadratic formula comes from. For plain solving, the formula below is usually faster.
Method 3 — The Quadratic Formula
WORKS FOR EVERY QUADRATICThe quadratic formula solves any equation in standard form, no guessing required. It comes directly from completing the square on the general form ax² + bx + c = 0.
Step-by-step example: 2x² + 4x − 6 = 0
- Identify the coefficients: a = 2, b = 4, c = −6.
- Compute the discriminant: b² − 4ac = 16 − 4(2)(−6) = 16 + 48 = 64.
- Substitute into the formula: x = (−4 ± √64) / 4 = (−4 ± 8) / 4.
- Split the ± : x = (−4 + 8)/4 = 1 or x = (−4 − 8)/4 = −3.
- The roots are x = 1 and x = −3.
Try the interactive parabola below: adjust a, b, and c and watch the roots, the vertex, and the discriminant update live.
When the discriminant turns negative, the parabola no longer touches the x-axis — the roots become complex.
The Discriminant: What b² − 4ac Tells You
The expression under the square root in the quadratic formula is called the discriminant. It tells you how many real roots exist before you calculate them.
| Discriminant (Δ = b² − 4ac) | Roots | Graph behaviour |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | One real root (repeated) | Parabola touches x-axis at the vertex |
| Δ < 0 | No real roots (two complex) | Parabola never touches x-axis |
All Three Methods, Side by Side
Here's the whole picture in one place — what each method is best at, and when to reach for it.
| Method | Best for | Works always? | Speed |
|---|---|---|---|
| Factoring | Simple integer roots | No | Fastest when it works |
| Completing the square | Vertex form, perfect squares, deriving the formula | Yes | Medium |
| Quadratic formula | Any quadratic, especially messy or irrational roots | Yes | Medium — always reliable |
Where You'll See Quadratic Equations in Real Life
Quadratics aren't just a textbook exercise — they model anything that speeds up, slows down, or curves.
- Projectile motion: the height of a thrown ball over time follows h = −5t² + v₀t + h₀. Solving for t when h = 0 tells you when it lands.
- Area problems: a garden with length 3 m longer than its width and an area of 40 m² gives w(w + 3) = 40, a quadratic.
- Profit maximization: revenue minus cost often produces a downward parabola — the vertex is the maximum profit.
- Braking distance: a car's stopping distance grows quadratically with speed, which is why doubling your speed more than doubles how far you need to stop.
Quadratics model constant acceleration perfectly. When acceleration itself changes — or when you're dealing with more than one variable at higher powers — you move into higher-degree polynomials or calculus. But for a huge range of everyday problems, the quadratic is the right tool.