Answer:
m = -(2g - 4xy)/2x²w⁴ + √((2g - 4xy)² - 4*(x²w⁴)*(4y²w⁴ - w))/2x²w⁴
Step-by-step explanation:
2y= xm + √(w + 2gm) ÷ w^2
multiply all through by w²
2yw² = xm(w²) + √(w + 2gm)
rearranging
√(w + 2gm) = 2yw² - xmw²
square both sides
w + 2gm = (2yw² - xmw²)²
w + 2gm = (2yw² - xmw²)*(2yw² - xmw²)
w + 2gm = 4y²w⁴ - 2yxmw⁴ - 2yxmw⁴ + x²m²w⁴
collecting like terms
(x²w⁴)m² + (2g - 4yxw⁴)m + (4y²w⁴ - w). = 0
This seems to have been transformed to a quadratic equation
so we solve the quadratic equation by formula taking m as the variable
For simplicity, put
x²w⁴ = a
2g - 4xy = b
4y²w⁴ - w = c
we have
am² + bm + c = 0
solving the quadratic by formula
(-b + √(b² - 4ac))/2a. (taking the positive part)
-b/2a +√(b² - 4ac)/2a = m
substituting back
x²w⁴ = a
2g - 4xy = b
4y²w⁴ - w = c
m = -(2g - 4xy)/2x²w⁴ + √((2g - 4xy)² - 4*(x²w⁴)*(4y²w⁴ - w))/2x²w⁴
