二次式

The path of a thrown ball

• Throw a ball and it traces a parabola 抛物线 — the same curve described by every quadratic 二次式 equation.
• Galileo proved this in 1638. Today, quadratics appear in physics, economics, engineering, and optimization.
• Mastering them means you can find the peak, the roots, and the shape of any parabola.

Completing the square 配方法

• A quadratic is $ax^2 + bx + c$ (with $a \neq 0$).
• Completing the square rewrites it as $a(x+p)^2 + q$.
• This reveals the vertex 顶点 (turning point) at $(-p,\ q)$.
• Example: $x^2 - 6x + 5 = (x-3)^2 - 9 + 5 = (x-3)^2 - 4$ → vertex $(3, -4)$.

Completing the square reveals the vertex directly: $y = (x-3)^2 - 4$ has its minimum at $(3, -4)$.

The discriminant 判别式

• The discriminant is $\Delta = b^2 - 4ac$. It counts the real roots:
  • $b^2 - 4ac > 0$ → two distinct real roots,
  • $b^2 - 4ac = 0$ → one repeated root,
  • $b^2 - 4ac < 0$ → no real roots.
• The quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

The sign of the discriminant fixes how many times the parabola meets the x-axis: two roots, one, or none

A suspension bridge: the main cable hangs in a parabola

Solving quadratics

• Factorising: fastest when it works. $x^2 - 5x + 6 = (x-2)(x-3) = 0 \Rightarrow x = 2$ or $x = 3$.
• Completing the square: always works, reveals the vertex.
• Quadratic formula: always works, use when factorising is hard.

Linked equations and modelling

• Quadratics model: projectile paths, profit curves, bridge arches, satellite dishes.
• The vertex gives the maximum (if $a < 0$) or minimum (if $a > 0$) value.
• Example: a profit function $P = -2x^2 + 40x - 150$. Maximum at $x = \dfrac{-40}{2(-2)} = 10$.
• You should solve quadratic equations and quadratic inequalities, and a pair of simultaneous equations (one linear, one quadratic).

You've got it