A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The squared term is what makes it quadratic — removing it would collapse the equation into a simple linear one. Quadratics appear everywhere: the arc of a cricket ball, the shape of a satellite dish, compound interest over two periods, and the dimensions of a room given its area. Knowing three reliable methods to solve them puts you in control of a surprisingly wide range of problems.
Step 1 — Write the Equation in Standard Form and Identify a, b, c
Before applying any method, rearrange the equation so that one side is zero. Take 2x² − 4x − 6 = 0 as the running example throughout this guide. Divide every term by 2 to keep numbers small:
x² − 2x − 3 = 0
Now read off the coefficients:
- a = 1 (coefficient of x²)
- b = −2 (coefficient of x)
- c = −3 (constant term)
Keep the signs attached to the numbers — a sign error here will carry through every subsequent step.
Step 2 — Compute the Discriminant
Before choosing a method, calculate the discriminant: Δ = b² − 4ac.
For our example: Δ = (−2)² − 4(1)(−3) = 4 + 12 = 16
| Discriminant | Solutions |
|---|---|
| Δ > 0 | Two distinct real solutions |
| Δ = 0 | One real solution (double root) |
| Δ < 0 | No real solutions (complex roots) |
A positive discriminant of 16 tells us immediately that x² − 2x − 3 = 0 has two real solutions. The square root of 16 is a whole number, which also hints that factoring will work cleanly here.
Method 1 — Quadratic Formula (Always Works)
The quadratic formula is the universal method. It works for every quadratic equation, even when the roots are irrational or complex.
x = [−b ± √(b² − 4ac)] / 2a
Substituting a = 1, b = −2, c = −3:
x = [−(−2) ± √16] / 2(1) = [2 ± 4] / 2
- x₁ = (2 + 4) / 2 = 3
- x₂ = (2 − 4) / 2 = −1
For equations with large or decimal coefficients, skip the arithmetic and use the Quadratic Formula Calculator — enter a, b, and c and it returns both roots, the discriminant, and the vertex instantly.
Method 2 — Factoring (Fastest When It Works)
Factoring is the quickest route when the roots happen to be integers or simple fractions. The goal is to rewrite ax² + bx + c as a product of two linear factors, then apply the zero-product property.
For x² − 2x − 3 = 0, find two numbers that:
- Multiply to c = −3
- Add to b = −2
Those numbers are −3 and +1 (−3 × 1 = −3; −3 + 1 = −2).
Write the factored form: (x − 3)(x + 1) = 0
Set each factor to zero:
- x − 3 = 0 → x = 3
- x + 1 = 0 → x = −1
Same answers, less arithmetic. If you spend more than thirty seconds hunting for the pair, stop and use the formula instead — not every quadratic factors over the integers.
Method 3 — Completing the Square
Completing the square transforms the equation into a perfect square on the left side, making the square root step trivial. It is the algebraic proof behind the quadratic formula and is most useful when you also need the vertex of the parabola.
Starting from x² − 2x − 3 = 0:
- Move the constant to the right: x² − 2x = 3
- Add (b/2)² = (−2/2)² = 1 to both sides: x² − 2x + 1 = 4
- The left side is now a perfect square: (x − 1)² = 4
- Take the square root of both sides: x − 1 = ±2
- Solve: x = 1 + 2 = 3 or x = 1 − 2 = −1
The vertex of the parabola is at (1, −4), which you can read directly from step 3: h = 1, and k = −4 (the negative of the right-hand side before taking the root, adjusted for a).
Real-World Example — Projectile Motion
A ball is thrown upward from a height of 48 feet with an initial velocity of 32 ft/s. Its height at time t seconds is:
h = −16t² + 32t + 48
When does it hit the ground? Set h = 0:
−16t² + 32t + 48 = 0
Divide by −16: t² − 2t − 3 = 0
This is exactly the same equation solved above. The roots are t = 3 and t = −1. Since negative time has no physical meaning, the ball hits the ground at t = 3 seconds.
For higher-degree polynomial equations in problems like this, the Polynomial Calculator can handle the expanded forms automatically.
Choosing the Right Method
| Situation | Recommended method |
|---|---|
| Coefficients are large or non-integer | Quadratic formula |
| Roots look like integers and ac is small | Try factoring first |
| You need the vertex or vertex form | Completing the square |
| You want an instant verified answer | Quadratic Formula Calculator |
Summary
Every quadratic equation in standard form ax² + bx + c = 0 can be solved with three techniques: the quadratic formula (always reliable), factoring (fast for clean integers), and completing the square (also reveals the vertex). The discriminant b² − 4ac tells you before you start whether to expect two real roots, one double root, or complex roots. For the worked example x² − 2x − 3 = 0 all three methods give x = 3 and x = −1, which confirms they are equivalent — choose whichever fits the numbers in front of you.