is a straight line through (1, 0), the gradient of which will vary as k changes.

If k > 0 then the straight line will have a negative gradient and cross the W-shaped graph once between x = 0 and x = 1, so x = -1 cannot be the largest of the three x-values satisfying the original equation.

If k < 0 then the straight line will have a positive gradient and can only cross the W-shaped graph when x > 1 (if at all). If the graphs do cross, then x = -1 cannot be the largest of the three x-values satisfying the original equation. If they do not cross, then x = -1 is the only x-value satisfying the original equation, contrary to the conditions given.

k cannot be zero, as then the only x value satisfying this equation is x = -1 again.

So there is no case where x = -1 can be the largest x-value.

This makes it easy to find k, as x = -1 must be one of the two least values, which have a sum of -3. Therefore x = -2 must satisfy the quartic equation, meaning that

Challenge 357:Equation Enigma