Year 10: Completing the square

Completing the square is a technique used to rewrite a quadratic expression in the form $x+a\mathrm{²}+b$, which is essential for solving quadratic equations and working with quadratic functions.

Steps:

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

Example:$x\mathrm{²}+6x+5$

1. $b=6$, $c=5a=(6/2)\mathrm{²}=9$

2. $x\mathrm{²}+6x+9+9-9=-9+9-9\Rightarrow (x+3)\mathrm{²}=-9$

3. $x+3$ = ±√$(-9)\Rightarrow x=-3$ ± 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.