Quadratic formula practice questions and worked solutions

Any quadratic can be written in the general form ax² + bx + c = 0, and the formula x = (−b ± √(b² − 4ac)) / 2a returns the values of x that satisfy it. Geometrically those values are where the parabola crosses the horizontal axis, which is why a quadratic can have two solutions, one, or none at all.

The formula is derived by completing the square on the general form, and it is worth working through that derivation once even though you will never use it again directly. Students who have seen where the formula comes from make fewer sign errors with it, because the structure stops being arbitrary symbols to memorise.

The expression under the square root, b² − 4ac, is called the discriminant, and it answers the question of how many real solutions exist before you do any of the arithmetic. A positive discriminant gives two distinct solutions, zero gives one repeated solution, and a negative value gives none in the real numbers.

Checking it first is a genuine time saver in an exam. If the discriminant is negative you can state that there are no real solutions and move on rather than grinding through a formula that cannot produce an answer, and if it turns out to be a perfect square you know the quadratic factorises and there was a faster route available.

The discriminant also appears in its own right in senior questions, particularly ones asking for the values of a parameter that make an equation have exactly one solution. Those questions are asking you to set the discriminant to zero and solve, which is only obvious once you think of it as a quantity rather than as part of a larger formula.

The dominant error is sign handling on c. In x² + 3x − 5 = 0 the value of c is negative five, so −4ac becomes positive twenty and the discriminant grows rather than shrinks. Writing a, b and c down explicitly as a labelled line before substituting costs five seconds and removes most of these mistakes.

The second common loss is dividing only part of the numerator by 2a. The entire expression −b ± √(b² − 4ac) sits over the denominator, not just the square root term, and rushed working that drops the bracket produces an answer that looks plausible and is wrong.

A third pattern appears when a is negative. Multiplying through by negative one first, so that a becomes positive, makes the arithmetic considerably safer and does not change the solutions. Examiners award method marks for a correct approach, so tidying the equation before you start is never wasted.

The formula always works, which is exactly why students reach for it when something quicker is available. If the discriminant is a perfect square the quadratic factorises over the integers, and factorising is usually faster and less error-prone than substituting three values into a formula with a square root in it.

A quick habit that works well under time pressure is to look for two numbers that multiply to give ac and add to give b. If they are obvious within a few seconds, factorise; if they are not, go straight to the formula rather than continuing to hunt. Senior assessment rewards choosing efficiently, not just arriving at the answer.