Problem (Interesting Equation)

Solve the equation

x^1/3 + (1 - x)^2/3 = 2

Solution

You might be tempted to plot the curve

y = x^1/3 + (1 - x)^1/3

on your graphing calculator or personal computer and see when the curve has a height of y = 2, but this strategy fails since the curve never has a height of 2. So, we try a different strategy and raise both sides of the equation to the 3rd power and using the identity

(a + b)³ = a³ + 3 a² b + 3 a b² + b³

to get (after organizing terms)

3 x^1/3 (1 - x)^1/3 [x^1/3 + (1 - x)^1/3] = 8 - 1

were we now make the observation that the quantity inside [...] is equal to 2 (the original equation), and hence we have

6 x^1/3 (1 - x)^1/3 = 7

or simply

(x (1 - x))^1/3 = 7/6

Raising both sides to the 3rd power again gives the simple quadratic equation

x² - x + (7/6)³ = 0

which we can easily solve and has two complex roots

The fact the equation has complex roots is why the graph of

y = x^1/3 + (1 - x)^1/3

in the xy-plane never attains a height of y = 2.

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Last modified on Wednesday, March 17, 1999