The zero product property tells us that if we multiply two quantities together and get zero, then one or both of these is equal to zero.
In symbols this means that if ab = 0 then a = 0, b = 0 or both.
This means that if we have a product equal to zero one of both of the factors (the things we are multiplying together) is equal to zero.
Of the equations given, the only one that has product (a few terms multiplied together) equal to zero is -(x-1)(x+9)=0 so this is the one that can most appropriately be solved by the zero product property.
We can re-write -(x-1)(x+9) = 0 as (-1)(x-1)(x+9) = 0 which means (x-1) or (x+9) or both equal zero. It should be clear that -1 cannot equal zero.
To solve we set x - 1 = 0 and obtain x = 1.
Next we set x + 9 = 0 and obtain x = -9
The solutions to the equation are x = -9 and x = 1.
We are, however, asked for two equations. Though none of the remaining equations has a product set equal to zero, we can re-write the equation
by factoring 3x from each term on the left-hand side. Doing so gives us
. Now we have a product equal to zero which means that one or both of the factors on the left-hand side equals zero.
That is, 3x equals 0, or x-2 equals 0 or both do. If 3x = 0 then x = 0. If x-2=0 then x=2. As such the solution to the equation is x = 0, x = 2.
