Given a convex

lens of focal length of (x+5) cm and a concave lens of focal length x cm.

The 2 lenses are placed 30 cm apart coaxially i.e along the same axis with the convex lens on the left while the concave lens is on the right. A light bulb is placed to the left of the convex lens at a distance of 10 cm. It is observed that light rays that emerges from the concave lens are parallel to each other. Find the value of x.

Since the light rays that emerges from the concave lens are parallel,the light rays that emerges from the convex lens converges at the focal point of the concave lens.

Therefore,

1/(x+5) = 1/10 + 1/(30-x)

x^2-45x+100=0

Solving for x, x=2.344 or 4.266

But,the answer given is x=2.655.

Part of the solution given involves the equation

1/(x+5) = 1/10 + 1/(30+x)

instead of 1/(x+5) = 1/10 + 1/(30-x)

They've added a negative sign to the focal length of the concave lens i.e. -x instead of x.

Is the solution correct?

I feel the solution is wrong because we shouldn't add a negative sign to x (the focal length of the concave lens) since

*x is a variable*.

If it's a variable shouldn't the negative sign take care of itself?

E.g. x^2+6x+5=0

solving, (x+1)(x+5)=0

x=-1,-5

But,if I know x is negative (as in the case of the focal length of the concave lens), I couldn't simply add a negative sign in front of x :

(-x)^2+6(-x)+5=0

x^2-6x+5=0

(x-1)(x-5)=0

x=1,5 (a different set of solution)

So is the solution correct?