This lesson builds on your previous knowledge by introducing two advanced topics: solving simultaneous equations with fractions and mastering the three key methods for solving any quadratic equation.

1. Simultaneous Equations with Fractional Coefficients

You'll see equations like (1/2)x + (1/3)y = 20. The goal is to solve them just like regular simultaneous equations.

The Core Strategy: Eliminate the Fractions First! Working with fractions is slow and can lead to errors. The best strategy is to convert the equations into a form with only whole numbers.

How to Solve

1. Look at one equation at a time. Find the Least Common Multiple (LCM) of the denominators in that equation.

2. Multiply every term in the equation by the LCM. This will cancel out all the fractions.

3. Repeat for the second equation. You will now have two "normal" simultaneous equations with integer coefficients.

4. Solve these new equations using the methods you already know (elimination or substitution).

Example: For the equation (1/2)x + (1/3)y = 20, the LCM of 2 and 3 is 6. Multiply by 6: 6 (1/2)x + 6 (1/3)y = 6 * 20 3x + 2y = 120 ← Much easier to work with!

Exam Tip: Always double-check your multiplication. A small error in this first step will make the rest of your calculation incorrect.

2. Solving Quadratic Equations (ax² + bx + c = 0)

You need to be an expert in three methods. The method you choose depends on the question and the equation itself.

Method 1: Solving by Factoring

This is the fastest method, so you should always try it first.

1. Set the equation to zero: Make sure all terms are on one side (ax² + bx + c = 0).

2. Factor the expression: Find two numbers that multiply to give ac and add to give b.

3. Create two brackets: e.g., (x - p)(x - q) = 0.

4. Solve each bracket: Set each bracket equal to zero to find the two possible values for x.

When to use: Use this when you can spot the factors quickly. Perfect for equations like x² - 5x + 6 = 0.

Method 2: Solving by Completing the Square

This method is required if the question specifically asks for it, or if factoring is difficult.

1. Isolate x terms: Move the constant term (c) to the right side of the equation.

2. Find the "magic number": Take the coefficient of the x term (b), divide it by 2, and then square the result. (b/2)².

3. Add this number to both sides of the equation.

4. Factor the left side: The left side will now be a perfect square, e.g., (x + b/2)².

5. Solve for x: Take the square root of both sides (remembering the ± sign) and solve.

Common Mistake: Forgetting the ± when you take the square root. A quadratic equation usually has two solutions, and forgetting this will cause you to lose one of them.

Method 3: The Quadratic Formula

This is the universal method. It works for every quadratic equation, even when the others don't. You must memorize it perfectly.

The Formula: x = [-b ± √(b² - 4ac)] / 2a

How to use it:

1. Identify a, b, and c from your equation ax² + bx + c = 0. Be very careful with negative signs!

2. Substitute these values directly into the formula.

3. Calculate the value inside the square root (b² - 4ac) first.

4. Calculate the two possible values for x, one using + and one using -.

Exam Tip: The formula is your most reliable tool. If you're struggling to factor an equation, don't waste time. Switch to the formula immediately. Pay close attention to signs, especially -b (if b is -7, then -b is +7) and -4ac (if c is negative, this part becomes positive).