Why is this way of calculating the image distance wrong?

Consider a point source of light placed $40$ cm to the right of a thin converging lens of focal length $15$ cm. We wish to find the distance of its image from the lens.

We can use the lens formula:

$$frac{1}{f}=frac{1}{v}-frac{1}{u}$$

Putting $f=+15$ cm and $u=-40$ cm,

$$v=frac{-15cdot 40}{-25}text{ cm}=+24 text{ cm}$$

This implies that the image will be formed $24$ cm to the left of the lens.

My question: we have defined a coordinate system where the center of the lens is the origin, and the object is on the negative x-axis. What happens if we take its distance from the lens to be positive, i.e., $+40$ cm?

My intuition is that the focal length will remain positive, i.e., $+15$ cm, for this reason: there is no change in the value of the focal length derived from the lens maker’s formula, which is:

$$frac{1}{f}=(mu-1)left(frac{1}{R_1}-frac{1}{R_2}right)$$

When we take the object to be on the negative x-axis, $R_2>0$ and $R_1<0$. When we take the object to be on the positive x-axis, $R_2<0$ and $R_1>0$. So it is obvious, from the formula at least, that the focal length doesn’t change whether we take the object distance to be positive or negative.

But how can this be? Taking the positive object distance and the same focal length,

$$v=frac{15cdot 40}{55} text{ cm}=10.9text{ cm}$$

This, contrary to our previous calculation, implies that the image will form $10.9$ cm to the right of the lens.

My textbook says that only the former value is correct, which is to say, you have to take the object distance to be negative. Why is this so?