2.2.1 Solving Quadratic Equations

Master multiple methods for solving quadratic equations including factoring, completing the square, quadratic formula, and graphical methods.

定义

A quadratic equation is a polynomial equation of the second degree, which can be written in the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are real coefficients and \(a \neq 0\). Solving a quadratic equation means finding all values of \(x\) (called roots or solutions) that satisfy the equation. These solutions can be real or complex numbers. There are multiple methods to solve quadratic equations: (1) Factoring, which involves expressing the quadratic as a product of linear factors; (2) Completing the square, which transforms the equation into a perfect square trinomial; (3) The quadratic formula, which provides a direct algebraic solution; and (4) Graphical methods, which involve finding the x-intercepts of the parabola \(y = ax^2 + bx + c\). The discriminant \(\Delta = b^2 - 4ac\) determines the nature of the roots: if \(\Delta > 0\), there are two distinct real roots; if \(\Delta = 0\), there is one repeated real root; if \(\Delta < 0\), there are two complex conjugate roots.

核心公式

• \(ax^2 + bx + c = 0\)
• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• \(\Delta = b^2 - 4ac\)
• \(x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4} + c\)
• \(a(x - r_1)(x - r_2) = 0\), where \(r_1\) and \(r_2\) are the roots

易错点

• ⚠️ Forgetting to check that \(a \neq 0\) before applying the quadratic formula, or incorrectly identifying coefficients when the equation is not in standard form (e.g., missing the negative sign in front of \(b\) in the formula)
• ⚠️ Making arithmetic errors when calculating the discriminant, especially with negative coefficients, leading to incorrect conclusions about the number and type of roots
• ⚠️ When factoring, finding factors of \(ac\) that sum to \(b\) but forgetting to properly factor out the leading coefficient \(a\) when \(a \neq 1\), or incorrectly splitting the middle term
• ⚠️ Misinterpreting the graphical method by confusing the vertex, axis of symmetry, or incorrectly reading the x-intercepts from a graph, or failing to recognize that the parabola may not intersect the x-axis (indicating complex roots)