Quadratic polynomial
Also known as: quadratic, second-degree polynomial
A quadratic polynomial is a polynomial of degree 2, written in standard form as ax² + bx + c with a ≠ 0. Its graph is a parabola, and it has at most two real roots.
A quadratic polynomial is a polynomial whose highest power of the variable is 2. In standard form it is written ax² + bx + c, where a, b, and c are constants and a ≠ 0 — if a were zero, the x² term would vanish and the expression would be linear, not quadratic. Examples include x² − 5x + 6, 3x², and −2x² + 7.
Setting a quadratic polynomial equal to zero produces a quadratic equation, which has at most two real solutions, called roots or zeros. You can find them by factoring (x² − 5x + 6 = (x − 2)(x − 3), giving roots 2 and 3), by completing the square, or with the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The discriminant, b² − 4ac, tells you how many real roots exist: two if it is positive, one repeated root if it is zero, and none if it is negative.
Graphically, a quadratic function y = ax² + bx + c is a parabola — opening upward when a > 0 and downward when a < 0, with its turning point (vertex) at x = −b / (2a). The real roots, when they exist, are exactly where the parabola crosses the x-axis. Two other useful shortcuts: the roots sum to −b/a and multiply to c/a.
Quadratics are a pillar of the SAT's Advanced Math domain, which tests factoring, the quadratic formula, the discriminant, and moving between standard, factored, and vertex forms. Fluency with all three representations — equation, factored form, and graph — is what the hardest questions demand.
Key takeaways
- A quadratic polynomial has degree 2: ax² + bx + c with a ≠ 0.
- It has at most two real roots, found by factoring, completing the square, or the quadratic formula.
- The discriminant b² − 4ac determines whether there are two, one, or no real roots.
- The graph is a parabola with vertex at x = −b / (2a), opening up if a > 0 and down if a < 0.
- The SAT tests quadratics heavily, including converting between standard, factored, and vertex forms.