Differentiate the quartic: f'(x)=4x³-15x²+9 where f(x)=x⁴-5x³+9x+3
Starting with x₀=5 apply Newton’s method:
x₁=x₀-f(x₀)/f'(x₀)
x₁=5-f(5)/f'(5)=5-48/134=4.641791
x₂=4.537543...
x₃=4.528973...
x₄=4.528917...
x₅=4.52891795...
So to 6 dec places x=4.528918