Year 10: Completing the square

Completing the square is a technique used to rewrite a quadratic expression in the form x + a² + b, which is essential for solving quadratic equations and working with quadratic functions.

Steps:

1. Start with a general quadratic expression: x² + bx + c
2. Move the constant term (c) to the right side of the equation: x² + bx = -c
3. Find the value of 'a' that, when added to both sides, creates a perfect square trinomial. a = (b/2)²
4. Add 'a' to both sides of the equation: x² + bx + a + a = -c + a + a
5. Factor the left side: (x + a)² = -c + a + a
6. Take the square root of both sides: x + a = ±√(-c + a + a)
7. Solve for x: x = -a ± √(-c + a + a)

Example: x² + 6x + 5

1. b = 6, c = 5 a = (6/2)² = 9

2. x² + 6x + 9 + 9 - 9 = -9 + 9 - 9 => (x + 3)² = -9

3. x + 3 = ±√(-9) => x = -3 ± 3i (Note: This involves imaginary numbers)

Key Points:

• Completing the square is most useful for solving quadratic equations.
• It is also used to rewrite quadratic expressions in vertex form.
• Remember to consider the constant term and add 'a' to both sides.