64.5k views
1 vote
Solve the equation without using a calculator


\boldsymbol{4x√(2x-x^2) =2x-1}

Solve the equation without using a calculator \boldsymbol{4x√(2x-x^2) =2x-1}-example-1
User Chackle
by
8.4k points

1 Answer

5 votes

Answer:

x = 1.92888 (5 d.p.)

Explanation:

Given equation:


4x√(2x-x^2)=2x-1

Square both sides:


\implies (4x√(2x-x^2))^2=(2x-1)^2

Simplify:


\implies16x^2(2x-x^2)=4x^2-4x+1


\implies32x^3-16x^4=4x^2-4x+1


\implies -16x^4+32x^3-4x^2+4x-1=0

Solve using the Newton-Rhapson method.


\boxed{\begin{minipage}{8 cm}\underline{The Newton-Rhapson iteration for solving f$(x) = 0$}\\\\$x_(n+1)=x_n-\frac{\text{f}\left(x_n\right)}{\text{f}\:'\left(x_n\right)}$\\\end{minipage}}

If f(x) = 0 then:


\text{f}(x)= -16x^4+32x^3-4x^2+4x-1

Differentiate f(x) to find f'(x):


\implies \text{f}\:'(x)=-64x^3+96x^2-8x+4

This means the iteration formula is:


x_(n+1)=x_n-\frac{-16{x_n}^4+32{x_n}^3-4{x_n}^2+4{x_n}-1}{-64{x_n}^3+96{x_n}^2-8{x_n}+4}

Let x₀ = 2.

Substitute this into the formula to find x₁:


\begin{aligned} \implies x_1&=2-(-16(2)^4+32(2)^3-4(2)^2+4(2)-1)/(-64(2)^3+96(2)^2-8(2)+4)\\ & =2-(-9)/(-140)\\&=1.935714286\end{aligned}

Substitute x₁ into the iteration formula to find x₂:


\implies x_2=1.928947473

Repeat until the solution is found:


\implies x_3=1.928876009


\implies x_4=1.928876002


\implies x_5=1.928876002

Therefore, the solution to the given equation is x = 1.92888 (5 d.p.).

User Thomas Martinez
by
7.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories