Solving Quadratics by Factorising

• Quadratic equation: A quadratic is any equation that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$. This standard form ensures the expression can be manipulated using algebraic methods designed for second‑degree polynomials.

• Factorising a quadratic: This means rewriting $ax^2 + bx + c$ as a product of two linear expressions such as $(px + q)(rx + s)$. Factorisation makes the structure of the equation more transparent and turns the solving process into two simpler linear equations.

• Zero‑product property: If $(A)(B) = 0$, then $A = 0$ or $B = 0$. This principle is the foundation of solving quadratics by factorisation because it allows you to split one equation into two solvable linear equations.

• Equivalence through rearrangement: A quadratic must be rearranged so that zero is on one side, because the zero‑product property only applies when the entire expression equals zero. This ensures that the factorised form correctly represents all the equation’s solutions.

• Expression structure: The product $(px + q)(rx + s)$ expands to $prx^2 + (ps + qr)x + qs$. Understanding this structure helps you recognise how coefficients determine possible factor pairs and guides efficient factor selection.

• Roots and factors relationship: If a quadratic factorises into $(x - r_1)(x - r_2)$, then $r_1$ and $r_2$ are the solutions. This relationship underlies why solutions appear as the opposite sign to numbers inside the factor brackets when the coefficient of $x$ is 1.

Comparing Types of Factorisation

• Simple monic quadratics ($a = 1$): These typically factorise into $(x + p)(x + q)$, where $p$ and $q$ are integers whose sum and product match the coefficients. These are usually the quickest to solve.

• Non‑monic quadratics ($a \neq 1$): These require more structured factorisation because the factors involve coefficients of $x$. In such cases, you cannot simply change signs in the brackets to read off the solution; each factor must be solved individually.

• Quadratics with a common factor: When all terms share a factor such as $x$, this factor must be kept rather than cancelled, because it represents a valid solution such as $x = 0$.

Key Idea: Cancelling a factor removes potential solutions and should be avoided unless solving rational equations, not quadratics.

• Losing the zero solution: Cancelling a factor like $x$ from both sides removes the root $x = 0$ from the solution set. This mistake happens because students incorrectly treat a quadratic like a rational equation.

• Incorrectly reading factors: In non‑monic quadratics such as $(ax + b)(cx + d)$, assuming solutions are simply the opposite of $b$ and $d$ leads to incorrect answers. Each factor must be solved explicitly to find the correct values.

• Sign errors in factor pairs: Students often misjudge sign combinations, especially when the middle coefficient is negative. Careful sign checking reduces incorrect factorisation attempts.

• Link to the quadratic formula: Factorising is essentially the inverse of expanding; when factorisation becomes difficult or impossible using integers, the quadratic formula provides a universal alternative. This connection illustrates how different methods complement each other.

• Graphical interpretation: The solutions found by factorising correspond to the x‑intercepts of the parabola $y = ax^2 + bx + c$. Understanding this link aids conceptual reasoning and supports strategic method selection.

• Preparation for algebraic manipulation: Factoring quadratics develops skills used in simplifying rational expressions, solving polynomial equations of higher degree, and performing partial fraction decomposition.