Let's start by considering whether a solution exists. There is only one solution to the equation in real numbers, and it lies between 0 and 10. This follows from two facts: a) both the left and right sides of the equation are monotonically increasing, with the left side growing faster. b) for x=0, the left side is smaller than the right side, while for x=10, the left side is larger. Let's transform the equation to the form: x^2(x-1) = 180 = 3^2 * 2^2 * 5 = 2^2 * = 3^2 * 20 = 6^2 * 5 and we represented 180 as all possible products of a square and a natural number. From this, we observe that x=6 satisfies the equation. Another way : 180 = x^2(x-1) < x^3 => x > 180^1/3 => x >= 6 => x^2 >= 36 => x-1
Let's start by considering whether a solution exists.
There is only one solution to the equation in real numbers, and it lies between 0 and 10.
This follows from two facts:
a) both the left and right sides of the equation are monotonically increasing, with the left side growing faster.
b) for x=0, the left side is smaller than the right side, while for x=10, the left side is larger.
Let's transform the equation to the form:
x^2(x-1) = 180 = 3^2 * 2^2 * 5 = 2^2 * = 3^2 * 20 = 6^2 * 5
and we represented 180 as all possible products of a square and a natural number. From this, we observe that x=6 satisfies the equation.
Another way :
180 = x^2(x-1) < x^3 => x > 180^1/3 => x >= 6 => x^2 >= 36 => x-1