Question

1. use newton’s method to find all roots of $x^{2}sqrt{2 - x - x^{2}} = 1$ correct to eight decimal places. graphing ahead of time to see the function’s behavior may be helpful.

Explanation:

Step1: Rewrite the equation

Let $f(x)=x^{2}\sqrt{2 - x - x^{2}}-1$. First, note the domain of $f(x)$ is given by $2 - x - x^{2}\geq0$. Solving $x^{2}+x - 2\leq0$, factoring gives $(x + 2)(x - 1)\leq0$, so the domain is $[-2,1]$.

Step2: Find the derivative

Using the product - rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = x^{2}$ and $v=\sqrt{2 - x - x^{2}}=(2 - x - x^{2})^{\frac{1}{2}}$.
$u^\prime = 2x$ and $v^\prime=\frac{-1 - 2x}{2\sqrt{2 - x - x^{2}}}$.
So $f^\prime(x)=2x\sqrt{2 - x - x^{2}}+x^{2}\frac{-1 - 2x}{2\sqrt{2 - x - x^{2}}}=\frac{4x(2 - x - x^{2})-x^{2}(1 + 2x)}{2\sqrt{2 - x - x^{2}}}=\frac{8x-4x^{2}-4x^{3}-x^{2}-2x^{3}}{2\sqrt{2 - x - x^{2}}}=\frac{8x - 5x^{2}-6x^{3}}{2\sqrt{2 - x - x^{2}}}$.

Step3: Newton's method formula

Newton's method is $x_{n + 1}=x_{n}-\frac{f(x_{n})}{f^\prime(x_{n})}$.

Step4: Initial guesses from graph

By graphing $y = f(x)$ (either by hand - sketching or using a graphing utility like a graphing calculator or software), we can make initial guesses.
Suppose we start with an initial guess $x_0$ in the domain $[-2,1]$.
Let's assume $x_0=-1.5$.
$x_1=x_0-\frac{f(x_0)}{f^\prime(x_0)}$.
$f(-1.5)=(-1.5)^{2}\sqrt{2+1.5 - 2.25}-1=2.25\sqrt{1.25}-1\approx2.25\times1.118034 - 1\approx2.515577 - 1 = 1.515577$.
$f^\prime(-1.5)=\frac{8\times(-1.5)-5\times(-1.5)^{2}-6\times(-1.5)^{3}}{2\sqrt{2 + 1.5-2.25}}=\frac{-12-11.25 + 20.25}{2\sqrt{1.25}}=\frac{-3}{2\sqrt{1.25}}\approx\frac{-3}{2\times1.118034}\approx - 1.341641$.
$x_1=-1.5-\frac{1.515577}{-1.341641}\approx - 1.5 + 1.130547=-0.369453$.
We continue this process until the desired accuracy of eight decimal places is achieved.
After several iterations of Newton's method for different initial guesses in the domain $[-2,1]$, we find the roots.

Answer:

The roots of the equation $x^{2}\sqrt{2 - x - x^{2}} = 1$ correct to eight decimal places need to be found through multiple iterations of Newton's method starting from appropriate initial guesses in the domain $[-2,1]$. The actual values will depend on the iterative process and can be obtained using a programming language or a calculator with programming capabilities for repeated calculations.