Yet, every time I deal with a problem like the one below, I'm surprised at how many students struggle to solve the equation, when it can in fact be solved either intuitively or by using regular operations on the equation.
0=4x^2
If you give a student 64=4x^2, they'll divide by 4, take the square root of both sides, and then give you the right answer (+/- 4). However, they look at 0=4x^2 as a completely different beast. They don't even know where to begin.
It's weird...
Perhaps this comes from confusion about dividing by zero. They've been told, you can't divide BY zero. They often mix up dividing one by zero with dividing zero by one. Therefore, in their heads they may be thinking, you can't divide by zero, when in fact the problem doesn't require dividing by zero, but dividing by 4.
They don't look at the intuitive solution either. They could all intuitively tell me that when I have two numbers and their product is zero, one of the numbers must be zero (Zero Product Property). However, they don't see that 4x^2 is just 4 times x times x. Therefore, the x must be zero in order for the product to be zero.
Overall, I feel like students have a very fragile understanding of what a product and it's factors are. That's what makes factoring polynomials so difficult too.
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