Solving Quadratic Equations by Factorising

• Quadratic Equation: A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term where the variable is squared. Its standard form is $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$. The coefficients $a, b, c$ determine the shape and position of the parabola when graphed.

• Factorising: Factorising is the algebraic process of rewriting a quadratic polynomial as a product of two or more simpler expressions, typically two linear binomials. For example, the quadratic expression $x^2 + 5x + 6$ can be factorised into $(x+2)(x+3)$, revealing its constituent linear components.

• Zero Product Property: This is the fundamental principle underpinning solving by factorising, stating that if the product of two or more real numbers is zero, then at least one of those numbers must be zero. Mathematically, if $A \cdot B = 0$, then either $A=0$ or $B=0$ (or both), which allows a quadratic equation to be broken down into simpler linear equations.

General Factorisation Method

• Step 1: Standard Form: The first step is to rearrange the given quadratic equation into its standard form, $ax^2 + bx + c = 0$. This may involve moving all terms to one side of the equation and combining like terms, ensuring that one side is precisely zero.

• Step 2: Factorise the Quadratic Expression: Next, factorise the quadratic expression $ax^2 + bx + c$ into a product of two linear binomials, such as $(px+q)(rx+s)$. This step often involves techniques like trial and error, grouping, or using specific factorisation patterns.

• Step 3: Apply Zero Product Property: Once factorised, apply the Zero Product Property by setting each of the linear factors equal to zero. For example, if $(px+q)(rx+s) = 0$, then you set $px+q=0$ and $rx+s=0$.

• Step 4: Solve Linear Equations: Finally, solve each of the resulting linear equations for $x$. Each linear equation will yield one of the two solutions (roots) for the original quadratic equation.

Handling Special Cases

• Common Factor 'x': If the quadratic equation has a common factor of $x$ (e.g., $x^2 - 4x = 0$), factor it out to get $x(x-4)=0$. Then, apply the Zero Product Property by setting $x=0$ and $x-4=0$, yielding solutions $x=0$ and $x=4$. It is crucial not to divide by $x$ at the beginning, as this would lose the $x=0$ solution.

• Factors with Coefficients: When the linear factors contain coefficients for $x$ (e.g., $(2x-3)(x+5)=0$), the process remains the same but requires an additional step in solving the linear equations. For instance, $2x-3=0$ leads to $2x=3$, so $x=3/2$, and $x+5=0$ leads to $x=-5$.

• Factorising an Expression vs. Solving an Equation: Factorising an expression, such as $x^2+5x+6$, results in an equivalent expression like $(x+2)(x+3)$. In contrast, solving a quadratic equation, like $x^2+5x+6=0$, involves finding the specific numerical values of $x$ that satisfy the equation, which are $x=-2$ and $x=-3$. The key difference lies in the presence of the equality sign and the subsequent application of the Zero Product Property to find specific values.

• Factorising vs. Other Solving Methods: Factorising is distinct from methods like completing the square or using the quadratic formula because it relies on algebraic manipulation to break down the quadratic into linear factors. While factorising provides exact solutions and is often quicker for easily factorable quadratics, other methods are universally applicable, even when factorisation is difficult or impossible (e.g., for irrational or complex roots).

• Prioritise Standard Form: Always begin by ensuring the quadratic equation is in the standard form $ax^2 + bx + c = 0$. This is a non-negotiable first step, as attempting to factorise an equation not set to zero will lead to incorrect solutions and lost marks.

• Avoid Dividing by Variables: A critical rule is never to divide both sides of an equation by a variable (e.g., $x$) if that variable could be zero. Doing so will eliminate a potential solution, typically $x=0$, which is a common error in exams. Instead, factor out the common variable and apply the Zero Product Property.

• Verify Solutions: After finding the solutions, substitute them back into the original quadratic equation to check for correctness. This simple verification step can quickly catch arithmetic errors or mistakes in factorisation, ensuring accuracy in your final answers.

• Recognise Factorable Forms: Develop an eye for common factorisation patterns, such as difference of two squares ($a^2-b^2 = (a-b)(a+b)$) or perfect square trinomials. Recognising these patterns can significantly speed up the factorisation process during an exam.

• Incorrect Standard Form: A frequent mistake is attempting to factorise an equation that is not set to zero, such as $x^2+3x=10$. Students might incorrectly factor $x^2+3x$ and try to equate factors to 10, which is mathematically unsound because the Zero Product Property only applies when the product equals zero.

• Losing the $x=0$ Solution: When an equation like $x^2 - 5x = 0$ is encountered, a common error is to divide both sides by $x$ to get $x-5=0$, thus finding only $x=5$. This overlooks the valid solution $x=0$, which is correctly found by factorising $x(x-5)=0$ and setting each factor to zero.

• Sign Errors in Solving Factors: Students often make sign errors when solving the linear equations derived from the factors, especially with negative constants or coefficients. For example, solving $2x+5=0$ might incorrectly yield $x=5/2$ instead of $x=-5/2$, due to a failure to correctly transpose the constant term.

• Assuming All Quadratics are Factorable: Not all quadratic equations can be easily factorised into linear factors with rational coefficients. A misconception is to force factorisation even when it's not straightforward, leading to frustration or incorrect factors. In such cases, other methods like the quadratic formula are more appropriate.

• Graphing Quadratics: The solutions (roots) obtained by factorising a quadratic equation $ax^2 + bx + c = 0$ directly correspond to the x-intercepts of the parabola represented by the function $y = ax^2 + bx + c$. These are the points where the graph crosses or touches the x-axis, providing a visual interpretation of the algebraic solutions.

• Polynomial Roots: Factorising is a foundational technique that extends to finding roots of higher-degree polynomials. By breaking down complex polynomials into simpler linear or quadratic factors, the Zero Product Property can be applied repeatedly to find all the roots of the polynomial.

• Discriminant and Factorability: While not directly part of the factorisation process, the discriminant ($b^2-4ac$) from the quadratic formula can offer insight into whether a quadratic is factorable. If the discriminant is a perfect square (e.g., 0, 1, 4, 9), the quadratic can be factorised into linear factors with rational coefficients.