Take x squared minus 5x plus 6 equals zero. The answers are x equals 3 and x equals 2, and you can check them in seconds: 9 minus 15 plus 6 is zero, and 4 minus 10 plus 6 is zero. The quadratic formula finds those answers for any quadratic equation, even when the answers are fractions, decimals, or complex numbers that no amount of guessing would produce. This page walks through the formula, the discriminant that predicts the answer type before you solve, and the mistakes that cost students the most points.

Standard form first.

Every quadratic equation must be rearranged into standard form before the formula can touch it: a times x squared, plus b times x, plus c, equals zero. The three coefficients are simply the numbers in front of each piece. In 2x squared plus 3x minus 2 equals zero, a is 2, b is 3, and c is negative 2. The sign travels with the number — forgetting that c is negative 2 rather than positive 2 is the single most common error on this topic. One hard rule: a cannot be zero. If a is zero, the x-squared term vanishes and the equation is linear, not quadratic, solved by ordinary algebra instead. The sign of a also tells you which way the parabola opens: positive a opens upward like a smile, negative a opens downward like a frown.

The formula, piece by piece.

For a times x squared plus b times x plus c equals zero, with a not zero, x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a. The plus-or-minus is doing real work: it is why most quadratics have two answers. Work the example 2x squared plus 3x minus 2 equals zero. First the discriminant, b squared minus 4ac: 9 minus 4 times 2 times negative 2, which is 9 plus 16, which is 25. Then x equals negative 3 plus or minus the square root of 25, divided by 4 — that is negative 3 plus or minus 5, over 4. The plus branch gives 2 over 4, which is 0.5. The minus branch gives negative 8 over 4, which is negative 2. Two answers, both checkable: 2 times 0.25 plus 1.5 minus 2 is zero, and 2 times 4 minus 6 minus 2 is zero.

The discriminant decides the story.

The piece under the square root, b squared minus 4ac, is called the discriminant, and it tells you what kind of answers to expect before you finish solving. If it is positive, there are two distinct real roots, and the parabola crosses the horizontal axis twice. The textbook case is x squared minus 8x plus 12 equals zero: the discriminant is 64 minus 48, which is 16, and the roots are 6 and 2. If it is zero, there is exactly one repeated root, and the parabola just touches the axis at its vertex: x squared minus 2x plus 1 equals zero has discriminant 4 minus 4, which is zero, and the single root is 1. If it is negative, there are no real roots at all — the parabola never meets the axis — and the answers are two complex conjugates. The classic is x squared plus x plus 1 equals zero: the discriminant is 1 minus 4, which is negative 3, and the roots are negative 1 plus or minus i times the square root of 3, all divided by 2. Complex roots always come in conjugate pairs, which is why the plus-or-minus matters twice over.

The vertex and the axis of symmetry.

A parabola is symmetric, and its mirror line runs through the vertex. The vertex's x-coordinate is negative b divided by 2a — the same denominator as the formula, with the square-root part set to zero. For x squared minus 5x plus 6, the vertex sits at x equals 2.5, exactly halfway between the roots 2 and 3, and the function's value there is 6.25 minus 12.5 plus 6, which is negative 0.25. That point is the parabola's minimum, since a is positive. This symmetry is a free error check: if your two roots do not average to negative b over 2a, at least one of them is wrong. For 2x squared plus 3x minus 2, the roots 0.5 and negative 2 average to negative 0.75, and negative 3 over 4 is negative 0.75 — the check passes.

Factoring versus the formula.

Factoring is faster when it works and useless when it does not. It works when the roots are tidy integers: x squared minus 5x plus 6 factors as x minus 3 times x minus 2, because 3 and 2 multiply to 6 and add to 5. It fails on equations like 2x squared plus 3x minus 2, whose roots 0.5 and negative 2 are not both integers, and on anything with irrational or complex roots. The formula never fails. The practical rule students learn the hard way: try factoring for ten seconds; if the factor pair does not jump out, go straight to the formula instead of burning five minutes hunting. And the factored form is always available after solving: if the roots are r1 and r2, the equation factors as a times x minus r1 times x minus r2. For x squared minus 5x plus 6, that is x minus 3 times x minus 2, which expands back to the original — the round trip that proves the roots are right.

Where the formula comes from.

The formula is not magic; it is completing the square, done once and for all. Divide ax squared plus bx plus c equals zero by a, move the constant to the other side, then add the square of half the x-coefficient to both sides to create a perfect square. The left side becomes x plus b over 2a, squared; the right side simplifies to b squared minus 4ac, over 4a squared. Take square roots, isolate x, and the quadratic formula falls out. Knowing the derivation matters for one practical reason: it explains why the discriminant appears. The square root step is exactly where the equation either has real answers, one repeated answer, or none — the discriminant is the quantity whose square root you are taking.

Projectile motion: the classic word problem.

Quadratic equations run physics, and the standard classroom version is a thrown ball. Height in feet after t seconds is often negative 16t squared plus 48t plus 64 — the 16 comes from half of gravity's 32 feet per second squared. Setting height to zero asks when the ball lands: negative 16t squared plus 48t plus 64 equals zero. The discriminant is 2,304 plus 4,096, which is 6,400, whose square root is 80. The roots are negative 48 plus or minus 80, divided by negative 32: that gives negative 1 and 4. Time cannot be negative here, so the ball lands after 4 seconds. The negative root is not a mistake — it is the time the ball would have launched if the clock had started earlier — but the physical answer is 4. Whenever a word problem gives you two roots and only one makes sense, the other one is usually telling you something about the setup, not an error.

Sign errors: where everyone slips.

Almost every wrong answer on this topic traces to a sign, not to the formula. Three traps cover most of them. First, negative b: if b is negative 7, then negative b is positive 7, and writing negative 7 instead flips both roots. The equation 2x squared minus 7x plus 3 equals zero has discriminant 49 minus 24, which is 25, and roots 3 and 0.5 — both positive, which is your clue that negative b must have been positive 7. Second, the discriminant's minus 4ac: when c is negative, minus 4ac becomes a plus, which is why 2x squared plus 3x minus 2 has discriminant 25 rather than negative 7. Third, dividing by 2a rather than 2: the denominator is twice the whole coefficient a, so for a equals 2 the denominator is 4, not 2. Write each step on its own line and the sign errors mostly vanish.

Messy decimals are fine.

Real problems rarely have integer coefficients, and the formula does not care. Take 4.43x squared minus 10.93x plus 3.59 equals zero. The discriminant is 119.4649 minus 63.6148, which is about 55.85. The square root of 55.85 is about 7.4733. The roots are 10.93 plus or minus 7.4733, divided by 8.86: about 2.08 and about 0.39. Rounding to two decimal places is standard for applied problems, and the check still works approximately: plugging 2.08 back in gives a value within a hundredth of zero, which is the rounding error, not a wrong answer. For homework that demands exact form, leave the square root unsimplified instead of decimalizing — exact answers beat rounded ones wherever the grader accepts them.

Checking your answer.

Every root can be verified by substitution, and you should do it at least for the first root. For x squared minus 5x plus 6 with root 3: 9 minus 15 plus 6 is zero, confirmed. With root 2: 4 minus 10 plus 6 is zero, confirmed. The factored form gives a second check: expand x minus 3 times x minus 2 and confirm you recover x squared minus 5x plus 6. The vertex check gives a third: the roots must average to negative b over 2a. Use any one of these on a test and most careless errors surface before you hand the paper in. If a root fails substitution, the error is almost always a sign in the discriminant or in negative b — recheck those two places first.

When a equals zero.

If the x-squared coefficient is zero, the equation is not quadratic at all. The equation 0 times x squared plus 4x minus 8 equals zero is really 4x minus 8 equals zero, solved as x equals 2. The quadratic formula would divide by zero and break, which is the formula's way of telling you the problem changed type. The same thing happens in disguise when an equation needs rearranging first: 3x squared equals 12 is quadratic (move everything to one side: 3x squared minus 12 equals zero, roots 2 and negative 2), but 4x equals 8 was never quadratic. Always collect everything on one side, in standard form, before deciding which tool the problem needs.

With the pre-filled defaults — Coefficient a (x²) 1, Coefficient b (x) -5, Constant c 6 — the Quadratic Formula Calculator returns Discriminant (b² - 4ac) 1.0000, Root x₁ 3.000000, Root x₂ 2.000000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.