k=x*(a-x)^2. x=?

Step 1: Start with the equation

k = x(a - x)^2

Step 2: Expand \( (a - x)^2 \):

(a - x)^2 = a^2 - 2ax + x^2

Step 3: Substitute the expanded form back into the equation:

k = x(a^2 - 2ax + x^2)

Step 4: Simplify the equation:

k = xa^2 - 2ax^2 + x^3

Step 5: Rearrange the equation to form a cubic polynomial:

x^3 - 2ax^2 + a^2x - k = 0

Step 6: Solve the cubic equation \( x^3 - 2ax^2 + a^2x - k = 0 \). The solution gives:

x = a - \sqrt{\frac{a^2 - k}{3}}

or

x = a + \sqrt{\frac{a^2 - k}{3}}

Answer:

x = a - \sqrt{\frac{a^2 - k}{3}}

or

x = a + \sqrt{\frac{a^2 - k}{3}}