How To Solve A Quadratic Equation

A quadratic equation is one that can be written as ax^2 + bx + c = 0 with a not equal to zero. That last condition is not a technicality: if a were zero the squared term would vanish and the equation would be linear, with one root instead of two and none of the machinery below applying. Checking it takes a second and prevents nonsense downstream.

The discriminant D = b^2 - 4ac deserves its reputation as the most informative quantity in the problem. Its sign tells you how many real solutions exist before any square root is taken: positive means two distinct real roots, zero means exactly one repeated root where the parabola just touches the axis, and negative means no real roots because the curve sits entirely above or below it. Its magnitude also predicts how far apart the roots lie.

Square roots explain why the two roots come in a pair. The term sqrt(D) is added to -b in one case and subtracted in the other, so the two answers sit symmetrically on either side of -b / (2a). That midpoint is the x-coordinate of the vertex, which means the vertex always lies exactly halfway between the roots whenever they exist — a fact worth using as a check on any answer you compute.

There is also a structure worth memorising because it catches sign errors immediately. With D = 1 the roots come out as 3 and 2, their sum is 5 which equals -b/a, and their product is 6 which equals c/a. Those two identities hold for every quadratic, so recomputing them from your answers is a fast check that costs nothing and detects most slips in transcription.

Factoring is quicker when it works. Rewriting x^2 - 5x + 6 as (x - 2)(x - 3) makes the roots obvious, and it is the right first attempt for textbook problems with small integer coefficients. But factoring only helps when roots happen to be convenient rationals; the quadratic formula always works, including for irrational answers like the roots of x^2 - 2 = 0. Completing the square is the third route, and it is the one that derives the formula rather than merely using it.

A negative discriminant is not a failure. It means the parabola does not cross the horizontal axis in the real plane, which is often the physically meaningful answer: a projectile launched too low to reach the ceiling, a revenue curve that never breaks even. Insisting on a numeric root in that situation produces complex numbers that may be correct algebraically and meaningless contextually.

Real-world quadratics almost always carry units, and units are what let you reject impossible answers. Solving -4.9t^2 + 20t + 2 = 0 for a thrown object gives t = 4.1793 seconds and t = -0.0977 seconds. Both satisfy the equation exactly; only one describes a time after launch. Discarding the negative root is not a mathematical judgement but a modelling one, and it should be stated rather than silently applied.

Precision deserves a word because it is easy to lose. Taking the difference of two nearly equal numbers inside the discriminant can cancel significant digits — computing b^2 - 4ac when b^2 barely exceeds 4ac loses accuracy fast. Rewriting one root using the relationship product = c/a avoids that cancellation and is the standard remedy in numerical work, even though it is invisible in textbook arithmetic.

Finally, remember what the solutions represent. Geometrically they are the axis intercepts of a parabola. Algebraically they are the values making the expression zero. In a model they are thresholds — break-even prices, arrival times, break points where a quantity changes sign. Keeping that reading in view is what makes the numbers useful after you have computed them.