Solving Quadratic Equations: Three Methods Explained
A quadratic equation has the form ax² + bx + c = 0. It appears in physics (projectile motion), optimization problems, and geometry. There are three standard methods for solving it — choosing the right one depends on the specific equation and how quickly you want the answer.
Factoring
If the equation factors neatly, this is the fastest approach. For x² + 5x + 6 = 0, you look for two numbers that multiply to 6 and add to 5: (x + 2)(x + 3) = 0, giving x = −2 or x = −3. Factoring works best for simple equations with integer roots; irrational or complex roots require another method.
The quadratic formula
For any quadratic ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is the discriminant: positive → two real solutions, zero → one real solution, negative → two complex solutions. The quadratic calculator applies this formula to any equation you enter.
What the discriminant tells you
Even before solving, the discriminant tells you how many real solutions exist. In applied problems like physics (when does a ball hit the ground?), a negative discriminant means the scenario has no real-world solution — the ball never reaches that height. The calculator shows the discriminant as part of its output.