Quadratic Formula: Derivation, Discriminant & Uses

The quadratic formula is the universal solver for quadratic equations: for ax squared + bx + c = 0 with a not zero, the solutions are x = (-b plus or minus root of (b squared - 4ac)) / (2a). It is derived by completing the square on the general quadratic, and it works for every quadratic equation, factorisable or not.

This matters because many real-world quadratics have messy coefficients or irrational roots that no factorisation can find. The formula never gets stuck: substitute three numbers, simplify, and the roots come out, real or otherwise.

Key Concepts

• Derivation: divide ax squared + bx + c = 0 by a, complete the square, and solve. The formula is simply completing the square done once and for all.
• The discriminant: the part under the square root, D = b squared - 4ac, which decides how many real roots exist.
• Plus-or-minus: the symbol that produces the two roots, one with + root D and one with - root D.
• Simplification: always simplify root D and cancel common factors at the end, for example (4 plus or minus 2 root 2)/2 = 2 plus or minus root 2.

Formulas

• Quadratic formula: x = (-b plus or minus root of (b squared - 4ac)) / (2a), with a not zero. If D is negative there are no real roots.
• Root count from D: D positive gives two distinct real roots, D zero gives one repeated root x = -b/(2a), D negative gives no real roots.

Worked Example

Solve x squared - 4x + 2 = 0 using the formula.

1. Identify a = 1, b = -4, c = 2. Note that b is -4, sign included.
2. Discriminant: D = (-4) squared - 4(1)(2) = 16 - 8 = 8.
3. Substitute: x = (4 plus or minus root 8) / 2.
4. Simplify root 8 = 2 root 2, so x = (4 plus or minus 2 root 2)/2 = 2 plus or minus root 2.
5. Numerically, x is about 3.414 or about 0.586.

Check x = 2 + root 2: (2 + root 2) squared = 4 + 4 root 2 + 2 = 6 + 4 root 2; then 6 + 4 root 2 - 4(2 + root 2) + 2 = 6 + 4 root 2 - 8 - 4 root 2 + 2 = 0. Verified.

Common Misconceptions

• b is 4 in x squared - 4x + 2 = 0. Wrong. The formula needs b with its sign, so b = -4 and -b = 4. Sign slips here are the top exam error.
• The denominator is 2, not 2a. Wrong. It is 2a, so for 2x squared - 7x + 3 = 0 you divide by 4, not 2.
• Rounding early is fine. Wrong. Keep exact surd form like 2 plus or minus root 2 through the working and round only the final answer if asked.

Real-World Applications

• In physics, equations of motion with awkward numbers, such as air-resistance models, are solved with the formula.
• In engineering, circuit and structural equations rarely have neat integer coefficients, so the formula is the default tool.
• In finance, compound-interest and loan equations that reduce to quadratics are solved exactly this way.

To recap: read off a, b and c with their signs, compute the discriminant, substitute carefully into the formula, and simplify the surd. The formula is completing the square pre-packaged, so it inherits the same reliability.