Who is asking: Parent
Level: Secondary
Question:
x2 = x+2
My daughter came home with this equation today and although I know the answer is x=2, i have no way of proving it by showing any working out.
Hi,
I assume that you arrived at x = 2 by just looking at the problem and realizing that 22 = 4 and that 2 + 2 = 4. This is something that we try to get students to do, use their ingenuity and number sense. One drawback with this method however is that you may miss something. That is the situation here as x = -1 is also a solution.
The procedural way to approach this problem is to first move the x term and the constant term to the left of the equal sign
x2 -x -2 = 0
and then factor the left side. Hence I want to find numbers a and b so that
x2 -x -2 = (x + a)(x + b)
If you expand the right side you will see that the constant term is a 
b and hence a b = -2. If a and b are integers then they must be -1 and 2 or 1 and -2. Try them
(x - 1)(x + 2) = x2 + x - 2
But you need x2 -x -2, so try the other possibilities.
(x + 1)(x - 2) = x2 - x - 2
which is what you need. Thus
(x + 1)(x - 2) = x2 - x - 2 = 0
and you have two numbers (x + 1) and (x - 2) whose product is zero. Thus one of them must be zero. That is
x + 1 = 0 and hence x = -1
or
x - 2 = 0 and hence x = 2.
Penny