Algebra Study Guide

A quadratic equation is an equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers, and a is not equal to zero. The highest power of the variable (usually x) in a quadratic equation is 2. This 'squared' term is what makes it quadratic. If a were zero, the equation would become bx + c = 0, which is a linear equation, not quadratic. The solutions to a quadratic equation are also known as its roots. These roots represent the x-values where the graph of the corresponding quadratic function (y = ax² + bx + c) intersects the x-axis. A quadratic equation can have two distinct real roots, one repeated real root (meaning the graph touches the x-axis at exactly one point), or no real roots (the graph does not intersect the x-axis). For foundation level, we typically focus on methods that lead to real number solutions, often by factoring or simple rearrangement. For example, in an equation like x² - 9 = 0, we can rearrange to x² = 9 and then take the square root of both sides to find x = ±3. In an equation like x² + 5x + 6 = 0, we look for two numbers that multiply to 6 and add to 5, which are 2 and 3. This allows us to factorise as (x+2)(x+3) = 0, leading to solutions x = -2 or x = -3.

Quadratic equations are formulas of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$. While basic methods like factorisation or graphically estimating roots are useful, higher methods are crucial for solving more complex or non-factorable quadratics. The primary advanced methods include completing the square and using the quadratic formula. Completing the square transforms the equation into the form $(x+p)^2 = q$, which can then be solved by taking the square root of both sides. This method is especially useful for understanding the structure of quadratic functions and deriving the quadratic formula itself. The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, is a universal tool that directly provides the solutions (roots) for any quadratic equation, regardless of whether it can be factorised. Understanding the discriminant, which is the term $b^2 - 4ac$ within the quadratic formula, is also a higher method. The discriminant determines the nature and number of roots: if $b^2 - 4ac > 0$, there are two distinct real roots; if $b^2 - 4ac = 0$, there is one repeated real root; and if $b^2 - 4ac < 0$, there are no real roots (two complex conjugate roots). These methods are fundamental for a wide range of mathematical and scientific problems, including curve sketching, optimization, and modelling physical phenomena.