How to Solve a Quadratic Equation
A quadratic is any equation that can be written as ax squared plus bx plus c equals zero, with a not zero. The Quadratic Equation Solver solves it in that form.
The discriminant tells you the answer's shape first
Before solving, calculate b squared minus 4ac. That single number - the discriminant - tells you what kind of solution exists.
Positive: two distinct real solutions. The curve crosses the x-axis twice. Zero: one repeated solution. The curve just touches the axis. Negative: no real solutions. The curve never reaches the axis, and the answers are complex numbers.
Knowing this before you grind through the formula is genuinely useful, because it tells you immediately whether an answer you were expecting exists at all. If a physical problem gives a negative discriminant, the situation you described cannot happen.
Three methods, and when each is quickest
Factorising is fastest when it works. Find two numbers that multiply to give ac and add to give b. For x squared minus 5x plus 6, that is -2 and -3, so it factors to (x-2)(x-3) and the solutions are 2 and 3. Clean, and it only works when the roots are rational.
The formula always works: x equals minus b, plus or minus the square root of the discriminant, all over 2a. It needs no insight and no luck, which is why it is the one to memorise.
Completing the square is the slowest of the three and the most informative, because it rewrites the equation in a form that shows the vertex of the parabola directly. It is also where the formula comes from - the formula is completing the square done once, in general.
What the solutions mean
A quadratic describes a parabola, and the solutions are where it crosses zero.
That is why quadratics turn up constantly in physics: anything under constant acceleration follows one. Throw a ball and its height over time is a quadratic, so solving it for zero tells you when it lands. Negative discriminant means it never reaches that height at all.
They also appear in anything involving area with a constraint, in optimisation, and in finance wherever a rate appears squared.
The mistakes worth watching
Signs. The formula has minus b at the front, so a negative b becomes positive. This is the commonest error by a wide margin.
Forgetting to rearrange. The formula needs the equation equal to zero. An equation like x squared equals 5x minus 6 must become x squared minus 5x plus 6 equals 0 first.
Dividing by 2a, not just a. And when a is negative, the whole denominator is negative.
Losing a solution. There are two, and both may be valid - or one may be physically meaningless, like a negative length, in which case discard it deliberately rather than by accident. For the simpler case of one unknown and no square, the Linear Equation Solver is the right tool.
Common questions
What if a is zero?
Then it is linear, not quadratic, and the formula divides by zero.
Can both solutions be the same?
Yes, when the discriminant is zero. That is a repeated root.
What are complex solutions used for?
A great deal of engineering and physics. For a geometry problem, they usually mean no real answer exists.
Got an equation to solve?