If 1 – x = x2 and x > 0, find the value of x3.

To find the value of x³ given the equation 1 - x = x² and x > 0, we first need to solve for x.

Step 1: Rearrange the equation to form a quadratic equation. x² + x - 1 = 0

Step 2: Solve the quadratic equation. This can be done using the quadratic formula, which is x = [-b ± sqrt(b² - 4ac)] / 2a. In this equation, a = 1, b = 1, and c = -1.

x = [-1 ± sqrt((1)² - 41(-1))] / 2*1 x = [-1 ± sqrt(1 + 4)] / 2 x = [-1 ± sqrt(5)] / 2

Since x > 0, we only consider the positive root.

x = [-1 + sqrt(5)] / 2 x = (sqrt(5) - 1) / 2

Step 3: Substitute x into x³ to find the value of x³.

x³ = [(sqrt(5) - 1) / 2]³ x³ = [(sqrt(5) - 1)³ / 8]

So, the value of x³ when 1 - x = x² and x > 0 is [(sqrt(5) - 1)³ / 8].