Expert: Josh - 2/1/2011

Question
Find the solutions of (ax+b)(cx+d)=0

Answer
Hi Jake,

We treat this question the same way as we would with numbers. Imagine you are looking at the product of two numbers, something like P times Q. If P*Q is zero, then one of three things must be true.

Either "P is zero",
OR "Q is zero",
OR "P and Q are both zero".

--This is due to the fact that any number multiplied by zero is zero--

How can we apply the same reasoning to this question. Well, we can simply call (ax+b) P, and treat (cx+d) as Q. Formally, we let P=ax+b, Q=cx+d. Since (ax+b)(cx+d)=0, we have P*Q=0.

To cover all possible solutions, we consider the three cases mentioned before.

Case 1: If P=0, then, by definition ax+b=0.
Subtracting b from both sides of the equation, we get ax=-b.
Dividing both sides by a, we get x=-b/a (as one of two possible solutions)

Case 2: If Q=0, then, cx+d=0.
Subtracting d from both sides of the equation, we get cx=-d.
Dividing both sides by c, we get x=-d/c (as the other possible solution)

Case 3: is a combination of (1) and (2).
So, the solutions of (ax+b)*(cx+d)=0 are x=-b/a and x=-d/c.