Completing the Square and Advanced Algebraic Fractions

TL;DR

Completing the square helps you rewrite quadratic expressions into a form that makes finding their turning point easy, and it's super useful for solving quadratic equations. Algebraic fractions are just like regular fractions but with letters, and you add, subtract, multiply, and divide them using similar rules. Together, these skills let you tackle more complex problems, especially when quadratics are involved in fractions.

1. The Mental Model

Think of completing the square as tidying up a messy quadratic expression into a neat little package that clearly shows its minimum or maximum point. Algebraic fractions are like puzzles where you need to find common denominators or factorise to simplify, just like you would with numbers.

2. The Core Material

Completing the Square

This is a powerful technique for rewriting a quadratic expression, $ax^2 + bx + c$, into the form $a(x+p)^2 + q$. This new form immediately tells you the turning point (vertex) of the parabola, which is at $(-p, q)$.

Let's start with the simplest case: quadratics where $a=1$, like $x^2 + bx + c$.

1. Halve the coefficient of x: Take the $b$ term, divide it by 2, and put it inside the bracket with $x$: $(x + \frac{b}{2})^2$.
2. Expand and adjust: If you expand $(x + \frac{b}{2})^2$, you get $x^2 + bx + (\frac{b}{2})^2$. Notice it has an extra constant term, $(\frac{b}{2})^2$, that wasn't in your original $x^2 + bx$ part.
3. Subtract the extra term: To balance this, you need to subtract that extra term immediately after the bracket: $(x + \frac{b}{2})^2 - (\frac{b}{2})^2$.
4. Add the original constant: Finally, bring back the original $c$ term: $(x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c$.

Example: Complete the square for $x^2 + 6x + 5$.

1. Half of $6$ is $3$. So we start with $(x+3)^2$.
2. Expand $(x+3)^2 = x^2 + 6x + 9$. We have an extra $9$.
3. Subtract the $9$: $(x+3)^2 - 9$.
4. Add the original $5$: $(x+3)^2 - 9 + 5 = (x+3)^2 - 4$.
   The turning point is $(-3, -4)$.

What if $a \neq 1$?

If you have something like $2x^2 + 8x + 10$:

1. Factor out $a$ from the $x^2$ and $x$ terms only: $2(x^2 + 4x) + 10$.
2. Complete the square inside the bracket for $x^2 + 4x$ (as shown above):
   $x^2 + 4x = (x+2)^2 - 2^2 = (x+2)^2 - 4$.
3. Substitute back: $2[(x+2)^2 - 4] + 10$.
4. Expand and simplify: $2(x+2)^2 - 8 + 10 = 2(x+2)^2 + 2$.
   The turning point is $(-2, 2)$.

Advanced Algebraic Fractions

These are fractions where the numerator and/or denominator contain algebraic expressions. The rules are the same as numerical fractions:

• Adding/Subtracting: Find a common denominator. This often means multiplying the denominators together, or finding the lowest common multiple if possible.
  $\frac{A}{B} + \frac{C}{D} = \frac{AD + BC}{BD}$
• Multiplying: Multiply numerators together and denominators together.
  $\frac{A}{B} \times \frac{C}{D} = \frac{AC}{BD}$
• Dividing: Flip the second fraction and multiply.
  $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C} = \frac{AD}{BC}$
• Simplifying: Factorise the numerator and denominator and cancel out any common factors. This is crucial for "advanced" problems.

Here's how algebraic fractions often involve factoring:

graph TD
    A["Start with Algebraic Fraction"] --> B{Can you Factorise Numerator or Denominator?};
    B -- Yes --> C["Factorise all parts (e.g., difference of squares, quadratic factoring)"];
    B -- No --> D{Is it Add/Subtract or Multiply/Divide?};
    C --> E{Any Common Factors?};
    E -- Yes --> F["Cancel out common factors"];
    E -- No --> D;
    D -- Add/Subtract --> G["Find Common Denominator"];
    D -- Multiply/Divide --> H["Apply rules (Multiply numerators/denominators, or Flip & Multiply for Division)"];
    G --> I["Combine Numerators over Common Denominator"];
    H --> I;
    I --> J{Can you Factorise and Simplify again?};
    J -- Yes --> F;
    J -- No --> K["Final Simplified Fraction"];

Example (Simplifying): Simplify $\frac{x^2 - 4}{x^2 + x - 6}$.

1. Factorise the numerator (difference of two squares): $x^2 - 4 = (x-2)(x+2)$.
2. Factorise the denominator (quadratic): $x^2 + x - 6 = (x+3)(x-2)$.
3. The fraction becomes $\frac{(x-2)(x+2)}{(x+3)(x-2)}$.
4. Cancel the common factor $(x-2)$: $\frac{x+2}{x+3}$.

3. Worked Example

Let's combine these skills.
Simplify $\frac{x^2 + 8x + 15}{x^2 - 9} \div \frac{x^2 + 6x + 9}{2x - 6}$.

1. Flip the second fraction and multiply:
   $\frac{x^2 + 8x + 15}{x^2 - 9} \times \frac{2x - 6}{x^2 + 6x + 9}$

2. Factorise each part:

   • $x^2 + 8x + 15 = (x+3)(x+5)$
   • $x^2 - 9 = (x-3)(x+3)$ (difference of two squares)
   • $2x - 6 = 2(x-3)$ (factor out common number)
   • $x^2 + 6x + 9 = (x+3)(x+3)$ or $(x+3)^2$ (perfect square trinomial)

3. Substitute the factored forms back into the multiplication:
   $\frac{(x+3)(x+5)}{(x-3)(x+3)} \times \frac{2(x-3)}{(x+3)(x+3)}$

4. Cancel common factors in the numerator and denominator:

   • One $(x+3)$ from the top left cancels with one $(x+3)$ from the bottom left.
   • The $(x-3)$ from the bottom left cancels with the $(x-3)$ from the top right.
   • One $(x+3)$ from the top right cancels with one $(x+3)$ from the bottom right.

   What's left is:
   $\frac{(x+5)}{1} \times \frac{2}{(x+3)}$

5. Multiply the remaining terms:
   $\frac{2(x+5)}{x+3}$

4. Key Takeaways

• Completing the square rewrites $ax^2 + bx + c$ into $a(x+p)^2 + q$, making the turning point $(-p, q)$ obvious.
• When $a=1$, you half the $x$ coefficient, square it, and subtract it to balance the expression.
• When $a \ne 1$, factor $a$ out of the $x^2$ and $x$ terms first, complete the square inside, then re-distribute $a$.
• Algebraic fractions follow the same rules as numerical fractions: find common denominators for add/subtract, multiply across for multiply/divide.
• Factorising is absolutely essential for simplifying algebraic fractions and cancelling terms.
• Difference of two squares ($a^2 - b^2 = (a-b)(a+b)$) and quadratic factorisation are frequently used.

Common Mistakes to Avoid:
- Forgetting to subtract the extra term $(\frac{b}{2})^2$ when completing the square.
- Forgetting to multiply the constant $q$ by $a$ when $a \ne 1$ in the final step of completing the square.
- Trying to cancel terms in algebraic fractions before factorising them fully.
- Not flipping the second fraction when performing division.

5. Now Try It

Take 15 minutes to practise these skills. First, complete the square for $3x^2 - 12x + 5$. Then, simplify the algebraic fraction $\frac{x^2 - 7x + 10}{x^2 - 25} + \frac{2}{x+5}$. Success looks like correctly transforming the quadratic into $a(x+p)^2 + q$ form, finding the turning point, and simplifying the fraction to a single, combined fraction with its numerator and denominator fully factorised (if possible).