# Solution to a RMO algebra practice question

Refer to question posted on blog of Mar 13 2018, reproduced here for your convenience. Compare your solution with the one given here:

Question:

Show that the following expression: $[4-3x+\sqrt{16+9x^{2}-24x-x^{3}}]^{1/3}+[4-3x-\sqrt{16+9x^{2}-24x-x^{3}}]^{1/3}$ remains constant in the interval $0 \leq x \leq 1$. Find this constant value.

Solution/proof:

Let y equal the given expression for x in the prescribed interval. Then, taking cube of both sides, we write

$y^{3}=8-6x+3xy$

$y^{3}-8-3x(y-2)=0$

$(y-2)(y^{2}+2y+4-3x)=0$

The only real value of $y=2$, a constant. The roots of the quadratic equation for $0 are complex. It is easy to check that $y=2$ for both $x=0$ and 1.

Alternately, derive the square roots of the expression within the radical; you can use the method of undetermined coefficients for this.

Cheers,

Nalin Pithwa.

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