Solving Quadratic Equations

Definition: A quadratic equation is a second-degree polynomial equation in a single variable $x$, expressed in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$.

Roots and Solutions: The solutions to the equation are called the roots or zeros, representing the points where the corresponding quadratic graph $y = ax^2 + bx + c$ intersects the x-axis.

Rearrangement: Before applying any solving method, it is essential to rearrange the equation so that all terms are on one side, resulting in a zero on the other side. This allows for the application of the Zero Product Property during factorisation.

Core Principle: Factorisation relies on the Zero Product Property, which states that if the product of two factors is zero ($A \times B = 0$), then at least one of the factors must be zero ($A = 0$ or $B = 0$).

Application: This method is most efficient when the coefficients $a$, $b$, and $c$ are small integers that allow the expression to be easily split into two linear brackets, such as $(x - r_1)(x - r_2) = 0$.

Process: Once the quadratic is factorised, each linear factor is set to zero to find the individual roots. If a quadratic cannot be easily factorised, alternative methods like the quadratic formula must be used.

Concept: Completing the square transforms the quadratic from $ax^2 + bx + c = 0$ into the form $a(x + p)^2 + q = 0$. This isolates the squared term, making it possible to solve for $x$ using square roots.

Procedure: For $x^2 + bx + c = 0$, the expression is rewritten as $(x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c = 0$. If $a \neq 1$, the coefficient $a$ must be factored out of the $x^2$ and $x$ terms first.

Utility: Beyond solving, this form is highly useful for identifying the turning point (vertex) of the quadratic graph, which occurs at the coordinates $(-p, q)$.

The Formula: The quadratic formula provides a direct way to calculate roots using the coefficients: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Universality: Unlike factorisation, the quadratic formula works for every quadratic equation, including those with irrational or complex roots. It is the most reliable method when coefficients are large or decimals.

Exact Values: When using the formula, it is standard practice to keep answers in surd form (e.g., $2 + \sqrt{3}$) unless a decimal approximation is specifically requested, to maintain mathematical precision.

Definition: The discriminant, denoted by $\Delta$, is the expression found under the square root in the quadratic formula: $\Delta = b^2 - 4ac$.

Predicting Roots: The value of the discriminant determines the number and type of solutions. If $\Delta > 0$, there are two distinct real roots; if $\Delta = 0$, there is exactly one real root (a repeated root); if $\Delta < 0$, there are no real roots.

Graphical Link: Graphically, $\Delta > 0$ means the parabola crosses the x-axis twice, $\Delta = 0$ means it touches the x-axis at its vertex, and $\Delta < 0$ means the entire graph stays above or below the x-axis.

Identification: A "hidden" or "disguised" quadratic is an equation that does not look like a quadratic initially but follows the pattern $a[f(x)]^2 + b[f(x)] + c = 0$. Common examples involve powers of $x$ (like $x^4$ and $x^2$) or trigonometric functions.

Substitution Method: To solve these, a temporary variable is introduced, such as $u = f(x)$, to transform the equation into a standard quadratic $au^2 + bu + c = 0$.

Final Step: After solving for $u$, it is critical to substitute the original function back in ($f(x) = u$) to find the final values for $x$. This may result in more or fewer solutions than a standard quadratic.