Methods: Factorisation, Quadratic Formula, and Completing the Square

Introduction

Key Concepts

Factorisation

Factorisation is a method of solving quadratic equations by expressing them as the product of their factors. This approach simplifies the equation, making it easier to find the roots. The general form of a quadratic equation is: $$ ax^2 + bx + c = 0 $$ To factorise, we look for two binomials that multiply to give the original quadratic equation. For instance, consider the equation: $$ x^2 + 5x + 6 = 0 $$ We seek two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the x term). These numbers are 2 and 3. Therefore, the equation can be factorised as: $$ (x + 2)(x + 3) = 0 $$ Setting each factor equal to zero gives the solutions: $$ x = -2 \quad \text{and} \quad x = -3 $$>

Steps to Factorise a Quadratic Equation:

1. Ensure the equation is in the standard form $ax^2 + bx + c = 0$.
2. Identify two numbers that multiply to $ac$ and add to $b$.
3. Rewrite the middle term using these numbers.
4. Factor by grouping.
5. Simplify to find the values of $x$.

Example: Factorise and solve: $$ 2x^2 + 7x + 3 = 0 $$ Solution: Find two numbers that multiply to $2 \times 3 = 6$ and add to 7. These numbers are 6 and 1. Rewrite the equation: $$ 2x^2 + 6x + x + 3 = 0 $$ Factor by grouping: $$ 2x(x + 3) + 1(x + 3) = 0 $$ Factor out $(x + 3)$: $$ (x + 3)(2x + 1) = 0 $$ Set each factor to zero: $$ x + 3 = 0 \quad \Rightarrow \quad x = -3 $$ $$ 2x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{2} $$

Advantages of Factorisation:

• Simplifies the process of finding roots.
• Requires fewer computational steps compared to other methods.
• Enhances understanding of the equation's structure.

Limitations of Factorisation:

• Not always applicable, especially when the equation does not factor neatly.
• May require trial and error to identify suitable factors.

Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, regardless of whether it can be factorised easily. The formula is derived from the standard form of a quadratic equation: $$ ax^2 + bx + c = 0 $$ The solutions for $x$ are given by: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$>

Derivation of the Quadratic Formula: To derive the quadratic formula, we complete the square on the general quadratic equation: 1. Divide both sides by $a$ (assuming $a \neq 0$): $$ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \quad \Rightarrow \quad x^2 + \frac{b}{a}x = -\frac{c}{a} $$ 2. Add $\left(\frac{b}{2a}\right)^2$ to both sides: $$ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 $$ 3. Simplify: $$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} $$ 4. Take the square root of both sides: $$ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$ 5. Solve for $x$: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Applying the Quadratic Formula: Example: Solve the equation: $$ 3x^2 - 2x - 1 = 0 $$ Solution: Identify $a = 3$, $b = -2$, and $c = -1$. Plug these values into the quadratic formula: $$ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-1)}}{2(3)} = \frac{2 \pm \sqrt{4 + 12}}{6} = \frac{2 \pm \sqrt{16}}{6} = \frac{2 \pm 4}{6} $$ Thus, the solutions are: $$ x = \frac{6}{6} = 1 \quad \text{and} \quad x = \frac{-2}{6} = -\frac{1}{3} $$

Advantages of the Quadratic Formula:

• Applicable to all quadratic equations.
• Provides exact solutions, including irrational and complex roots.
• Offers a systematic approach, eliminating the guesswork involved in factorisation.

Limitations of the Quadratic Formula:

• Requires accurate calculation, especially under exam conditions.
• May be more time-consuming compared to factorisation for simpler equations.

Completing the Square

Completing the square is a method that transforms a quadratic equation into a perfect square trinomial, making it easier to solve. This technique is particularly useful when factorisation is challenging or when deriving the quadratic formula. The general approach involves manipulating the equation to form $(x + p)^2 = q$, from which the solutions for $x$ can be easily found.

Steps to Complete the Square:

1. Start with the standard form $ax^2 + bx + c = 0$.
2. Divide all terms by $a$ to simplify if $a \neq 1$.
3. Move the constant term to the other side of the equation.
4. Take half of the coefficient of $x$, square it, and add to both sides.
5. Factor the perfect square trinomial on the left side.
6. Take the square root of both sides and solve for $x$.

Example: Solve the equation: $$ 2x^2 + 8x + 5 = 0 $$ Solution: 1. Divide by 2: $$ x^2 + 4x + \frac{5}{2} = 0 \quad \Rightarrow \quad x^2 + 4x = -\frac{5}{2} $$ 2. Take half of 4, which is 2, and square it to get 4. Add 4 to both sides: $$ x^2 + 4x + 4 = -\frac{5}{2} + 4 \quad \Rightarrow \quad (x + 2)^2 = \frac{3}{2} $$ 3. Take the square root of both sides: $$ x + 2 = \pm \sqrt{\frac{3}{2}} = \pm \frac{\sqrt{6}}{2} $$ 4. Solve for $x$: $$ x = -2 \pm \frac{\sqrt{6}}{2} $$>

Advantages of Completing the Square:

• Provides a clear understanding of the properties of quadratic equations.
• Essential for deriving the quadratic formula.
• Useful in graphing quadratic functions by identifying the vertex.

Limitations of Completing the Square:

• Can be algebraically intensive, especially with large coefficients.
• May lead to fractions, complicating calculations under timed conditions.

Advanced Concepts

In-depth Theoretical Explanations

Delving deeper into quadratic equations, it's crucial to understand the discriminant, which plays a pivotal role in determining the nature of the roots. The discriminant is the expression under the square root in the quadratic formula: $$ D = b^2 - 4ac $$>

Role of the Discriminant:

• Positive Discriminant ($D > 0$): Indicates two distinct real roots.
• Zero Discriminant ($D = 0$): Implies one real repeated root.
• Negative Discriminant ($D < 0$): Suggests two complex conjugate roots.

Understanding the discriminant is essential not only for solving equations but also for graphing quadratic functions, as it influences the intersection points with the x-axis.

Graphical Interpretation: The graph of a quadratic equation $y = ax^2 + bx + c$ is a parabola. The discriminant determines the number and type of intersections with the x-axis:

• If $D > 0$, the parabola intersects the x-axis at two points.
• If $D = 0$, the parabola touches the x-axis at the vertex.
• If $D < 0$, the parabola does not intersect the x-axis.

Derivation of the Quadratic Formula Using Completing the Square: Starting with the standard form: $$ ax^2 + bx + c = 0 $$> Divide by $a$: $$ x^2 + \frac{b}{a}x = -\frac{c}{a} $$> Complete the square: $$ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 $$> Factor and simplify: $$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} $$> Take the square root: $$ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$> Solve for $x$: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$> This derivation not only reinforces the completeness of the square method but also highlights its foundational role in algebraic problem-solving.

Complex Problem-Solving

Advanced problem-solving involves applying the three methods—factorisation, the quadratic formula, and completing the square—to complex scenarios, including word problems and equations with higher coefficients or requiring manipulation.

Example 1: A garden is designed in the shape of a rectangle. The area of the garden is 56 square meters, and the length is 2 meters more than twice the width. Find the dimensions of the garden. Solution: Let the width be $w$ meters. Then the length is $2w + 2$ meters. The area is: $$ w(2w + 2) = 56 \quad \Rightarrow \quad 2w^2 + 2w - 56 = 0 \quad \Rightarrow \quad w^2 + w - 28 = 0 $$> Apply the quadratic formula: $$ w = \frac{-1 \pm \sqrt{1 + 112}}{2} = \frac{-1 \pm \sqrt{113}}{2} $$> Since width cannot be negative: $$ w = \frac{-1 + \sqrt{113}}{2} \approx 5.19 \text{ meters} $$> Thus, the length is: $$ 2(5.19) + 2 = 12.38 \text{ meters} $$>

Example 2: Solve the equation using factorisation, the quadratic formula, and completing the square: $$ 5x^2 - 20x + 15 = 0 $$> Solution: First, simplify by dividing by 5: $$ x^2 - 4x + 3 = 0 $$> Factorisation: Find two numbers that multiply to 3 and add to -4: -1 and -3. $$ (x - 1)(x - 3) = 0 \quad \Rightarrow \quad x = 1, \quad x = 3 $$> Quadratic Formula: Here, $a = 1$, $b = -4$, $c = 3$. $$ x = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2} \quad \Rightarrow \quad x = 3, \quad x = 1 $$> Completing the Square: $$ x^2 - 4x = -3 $$> Take half of -4, which is -2, square it to get 4, and add to both sides: $$ x^2 - 4x + 4 = 1 \quad \Rightarrow \quad (x - 2)^2 = 1 $$> Take the square root: $$ x - 2 = \pm 1 \quad \Rightarrow \quad x = 3, \quad x = 1 $$> All methods yield the same solutions: $x = 1$ and $x = 3$.

Interdisciplinary Connections

Quadratic equations and their solutions are not confined to pure mathematics; they find applications across various disciplines, enhancing the analytical toolkit of students.

Physics: In kinematics, the equations of motion under uniform acceleration often lead to quadratic equations. For example, determining the time it takes for an object to reach a certain velocity involves solving: $$ v = u + at \quad \Rightarrow \quad at + u - v = 0 $$> If acceleration $a$ is not constant, the equation may become quadratic.

Engineering: Designing parabolic trajectories in projectiles or optimizing structures often requires solving quadratic equations to ensure precision and safety.

Economics: Quadratic models are used to determine profit maximization and cost minimization. For instance, calculating the price point that maximizes profit involves solving a quadratic equation derived from the revenue and cost functions.

Biology: Population models sometimes employ quadratic equations to predict growth rates under certain constraints, aiding in resource management and sustainability studies.

Computer Science: Algorithms for graphical computations, such as rendering quadratic Bézier curves, rely on solving quadratic equations to determine control points and ensure smoothness in graphics.

These interdisciplinary applications underscore the versatility and importance of mastering quadratic equations, fostering a holistic understanding that transcends mathematical theory.

Comparison Table

Summary and Key Takeaways