What makes an equation quadratic?
A quadratic equation can be written as ax² + bx + c = 0 with a ≠ 0. The highest variable power is two, so the equation can have as many as two roots over the complex numbers. Before solving, move every term to one side and combine like powers so the coefficients a, b, and c are unambiguous.
For 2x² + 7 = 5x, standard form is 2x² − 5x + 7 = 0. Here a = 2, b = −5, and c = 7. The negative sign belongs to b; losing it is one of the most common quadratic-formula mistakes.
Factoring or quadratic formula: which method should you use?
Factoring is efficient when the polynomial has recognizable rational factors. If ax² + bx + c = (px + q)(rx + s), the zero-product property sets each factor equal to zero. The square-root property is useful when the squared expression is already isolated. Completing the square is valuable for understanding vertex form.
The quadratic formula works for every quadratic equation with a nonzero a, including cases that do not factor over the rational numbers. A professional quadratic calculator can use a simple factorization when it makes the reasoning clearer and still analyze the discriminant to confirm the expected root type.
How to use the quadratic formula step by step
The formula is reliable only after the equation is in standard form. Keep coefficient signs attached when substituting, evaluate the discriminant before taking its square root, and simplify the two branches without converting exact radicals to decimals too early.
- Rewrite the equation as ax² + bx + c = 0 and confirm that a is not zero.
- Identify a, b, and c with their signs, then calculate D = b² − 4ac.
- Substitute into x = (−b ± √D)/(2a), keeping the numerator grouped.
- Simplify both the plus and minus branches and state the selected number domain.
- Substitute each candidate into the original equation or verify the factorization.
Worked example: x² − 5x + 6 = 0
The coefficients are a = 1, b = −5, and c = 6. The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1. Because 1 is a positive perfect square, the equation has two distinct rational roots.
Substitution into the formula gives x = (5 ± 1)/2. The plus branch gives x = 3 and the minus branch gives x = 2. Equivalently, the polynomial factors as (x − 2)(x − 3), so the zero-product property produces the same two roots.
What the discriminant tells you
When D > 0, there are two distinct real roots. If D is a perfect square and the coefficients are rational, those roots are rational; otherwise they are usually written with radicals. When D = 0, the two branches meet at one repeated real root. When D < 0, there are two complex conjugate roots and no real roots.
The Real or Complex domain control matters in the final case. Selecting real numbers should report that the equation has no real solution, while the complex domain can show values involving i, where i² = −1.
Check both roots and avoid formula errors
Always check both candidates. A quadratic solution is incomplete if one valid branch of ± is omitted. Substitute each root into the original equation, especially if terms were moved before standard form was created.
Common errors include using b instead of −b, forgetting to square all of b, placing only the radical over 2a, and rounding before the end. The fraction bar covers the entire numerator −b ± √D, which is why the calculator displays the formula as a true fraction.