Solving Quadratic Equations: Completing the Square

Quadratic equations are fundamental in algebra and serve as a basis for more advanced mathematical concepts.

Introduction to Quadratic Equations

• Quadratic Equation: Second-degree polynomial that forms a parabola.
• Standard Form: $ax^2 + bx + c = 0$
  • a: Coefficient of $x^2$
  • b: Coefficient of $x$
  • c: Constant term
• Important: a ≠ 0 for it to be quadratic.

Key Terms Defined

• Coefficients: Determine the shape and position of the parabola.
• Roots: Points where the parabola intersects the x-axis.
• Discriminant: $b^2 - 4ac$
  • Positive: Two distinct real roots
  • Zero: One real repeated root
  • Negative: No real roots
• Vertex: Represents the parabola's peak or trough.

Graphical Representation

• Quadratic equations graph as parabolas.
• Axis of Symmetry: Divides the parabola equally.
• Direction of Opening:
  • $a > 0$ - Opens upwards
  • $a < 0$ - Opens downwards

Introduction to Completing the Square

Completing the Square: This method transforms a quadratic equation into a form that eases the process of finding solutions for 'x'.

• Purpose: Transforms the equation into a perfect square trinomial.
• When to Use: Useful when factoring is challenging or when converting to vertex form.

Completing the Square Method

1. Initial Setup

   • Begin with the standard form: $ax^2 + bx + c = 0$.
   • Ensure that the coefficient of $x^2$ is 1.

2. Moving the Constant

   • Move the constant term to the opposite side: $ax^2 + bx = -c$.

3. Creating a Perfect Square

   • Add and subtract $\left(\frac{b}{2}\right)^2$ to the equation: $ax^2 + bx + \left(\frac{b}{2}\right)^2 = \left(\frac{b}{2}\right)^2 - c$

4. Factor as a Binomial Squared

   • Rewrite it as a binomial square: $(x + \frac{b}{2})^2 = \text{constant}$

5. Taking Square Roots

   • Calculate both $+\sqrt{\text{constant}}$ and $-\sqrt{\text{constant}}$.
   • Solve for 'x' in the equation $x + \frac{b}{2} = \pm\sqrt{\text{constant}}$.

Visual Guides

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Worked Examples

• Example 1: Solve $x^2 + 6x + 5 = 0$.

  • Step 1: Rearrange to $x^2 + 6x = -5$
  • Step 2: Half the coefficient of $x$ is 3, and $3^2 = 9$
  • Step 3: Add and subtract 9: $x^2 + 6x + 9 = -5 + 9$
  • Step 4: Factorise: $(x + 3)^2 = 4$
  • Step 5: Take square root: $x + 3 = \pm 2$
  • Step 6: Solve: $x = -3 \pm 2$, yielding $x = -1$ or $x = -5$

• Example 2: Solve $x^2 + x - 6 = 0$.

  • Step 1: Rearrange to $x^2 + x = 6$
  • Step 2: Half the coefficient of $x$ is 0.5, and $0.5^2 = 0.25$
  • Step 3: Add and subtract 0.25: $x^2 + x + 0.25 = 6 + 0.25$
  • Step 4: Factorise: $(x + 0.5)^2 = 6.25$
  • Step 5: Take square root: $x + 0.5 = \pm 2.5$
  • Step 6: Solve: $x = -0.5 \pm 2.5$, yielding $x = -3$ or $x = 2$

Vertex Form of Quadratic Equations

Vertex Form Expression

• Vertex Form: $y = a(x-h)^2 + k$.

• Purpose: Provides a clearer understanding of the parabola's characteristics, like its maximum or minimum point.

• Formula Elements:

  • $a$ (Direction and Width): Positive values mean the parabola opens upwards; negative values mean it opens downwards.
  • $(x-h)^2$ (Horizontal Shift): $h$ is the x-coordinate of the vertex.
  • $k$ (Vertical Shift): $k$ is the y-coordinate of the vertex.

Graphical Interpretation

• Visualising: Enables easy graphing of the parabola by recognising shifts and its direction.

Exam Tips

• Time Management: Start with simpler tasks to build confidence.
• Verify Solutions: Always substitute solutions back to confirm their correctness.
• Avoid Common Mistakes:
  • Fraction Errors: Double-check calculations involving complex fractions.
  • Sign Misuse: Be careful with signs during expansions and simplifications.