# Spurious solutions

1. Jun 20, 2008

### nokia8650

"Spurious solutions"

The question involves finding the solution to:

cos x + root (1-0.5sin2x) = 0

I can get the answer; first by taking one term to the other side, squaring, and then factorising.

0,180,45,225

However, the markscheme requir "Checking for spurious answers due to squaring"

Which results in the answers being 180 and 225. Can someone please explain what is meant by this and why, and how one would "check"?

Thanks

2. Jun 20, 2008

### dynamicsolo

When you squared both sides of the equation in order to complete the separation of x, this had the effect of eliminating the relevance of the sign of x. (In a sense, you are no longer solving your original equation.) When you have to do something like this, you need to try your solutions in the original equation to see which ones solve it, rather than your "constructed" equation.

Here's a simple example:

Solve x^2 = x , for real x.

Square both sides:

x^4 = x^2 ,

so (x^2) · [ (x^2) - 1 ] = 0 .

Thus, either x^2 = 0 , giving x = 0 , or

x^2 = 1 , giving x = +/-1 .

Plainly, x = 0 and x = 1 work in the original equation,
but x = -1 does not. This third result solves the "constructed" equation
x^4 = x^2 , but that is not strictly equivalent to the original equation x^2 = x (in fact, we are now really solving for x^2).
So we say that x = -1 is a "spurious solution".

(Of course, had you simply proceeded from (x^2) - x = x·(x - 1) = 0 , you would not have run into this. But this example simply demonstrates the issue -- and was what I came up with off the top of my head. In many situations, you have no option but to square both sides, introducing the difficulty I describe.)

Last edited: Jun 20, 2008
3. Jun 20, 2008

### nokia8650

ahh yes, thank you!

4. Jun 21, 2008

### PFStudent

Hey,

While I knew this before, I found your explanation very clear and refreshing. Thanks for the post dynamicsolo.

Thanks,

-PFStudent