Quadratic Equations and Applications

General Form of Quadratic Equations

• A quadratic equation in one variable $x$ is an equation that can be expressed in the form:   $ax^2 + bx + c = 0$   where $a$, $b$, and $c$ are constants and $a \neq 0$.

• A root or solution is a value of $x$ that satisfies the equation.

Factorisation Method

• Employs the zero product property: for any two real numbers $P$ and $Q$, if $P \times Q = 0$, then either $P = 0$ or Q = 0$.\n* Expressing an equation as (x - m)(x - n) = 0$gives solutions$x = m$or$x = n\n* Quadratic equations may result in two distinct roots or two **repeated roots** (equal roots).\n\n# Completing the Square Method\n\n* The algebraic process of **completing the square** transforms x^2 + bx$into a perfect square by adding$\left(\frac{b}{2}\right)^2:\n  x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2\n* An equation in the form (x + p)^2 = k$(where$k > 0) is solved by taking square roots on both sides:\n  x + p = \pm \sqrt{k} \implies x = -p \pm \sqrt{k}\n\n# Quadratic Formula Method\n\n* The roots of any quadratic equation ax^2 + bx + c = 0$(where$a eq 0) are determined by:\n  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n* If b^2 - 4ac < 0, the term under the square root is negative, meaning the equation has **no real roots** (no real solutions).\n\n# Solving Fractional Equations\n\n* Fractional equations contain variables in the denominator.\n* **Reduction**: Multiply both sides by the LCM of the denominators to convert the equation into standard quadratic form.\n* **Root Validation**: Derived roots must be checked against original denominators. Any root causing a division-by-zero error must be rejected, as division by zero is undefined.\n\n# Applications in Real-World Contexts\n\n* **Problem-Solving Framework**:\n 1. Identify an unknown quantity using a variable (e.g., x$or$v$$).

  2. Express all other given quantities in terms of that variable.

  3. Form and solve the resulting quadratic equation.

  4. Write a complete answer statement.

• Contextual Constraints: Always evaluate whether both roots are contextually valid. Reject negative or non-integer roots when the context demands positive real measures (e.g., lengths, geometric dimensions, speed).

Summary of Solving Methods