Use Newton's Method to approximate the zero(s) of the function. Continue the iterations until two suctessive approximations differ by less than 0.001 . Then find the zero(s) using a graphing utility and compare the results. (Round your answers to four dedmal places.) $$ f(x)=x-2 \sqrt{x+6} $$ Newton's method: $$ \quad x= $$ $$ \square $$ Graphing utility: $$ \quad x= $$ $$ \square $$

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We are given

$$
f(x)=x-2\sqrt{x+6}.
$$

Its zero satisfies

$$
x-2\sqrt{x+6}=0\quad \Longrightarrow \quad x=2\sqrt{x+6}.
$$

Because $$\sqrt{x+6}$$ is defined only for $$x\ge-6$$ and is nonnegative, any solution must have $$x\ge0$$.

The derivative is

$$
f'(x)=1-\frac{1}{\sqrt{x+6}}.
$$

We now apply Newton’s method with the iteration

$$
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
$$

A good starting guess is near the expected solution. (In fact, from algebra, squaring both sides gives
$$
x^2=4(x+6)\quad \Longrightarrow \quad x^2-4x-24=0,
$$
and the positive solution is $$x=2+2\sqrt7\approx7.29$$.) Let us choose $$x_0=7$$.

---

**Iteration 1:**

1. Compute $$\sqrt{x_0+6}=\sqrt{7+6}=\sqrt{13}\approx3.6056$$.

2. Evaluate

$$
   f(7)=7-2\sqrt{13}\approx7-2(3.6056)=7-7.2112\approx-0.2112.
   $$

3. Compute

$$
   f'(7)=1-\frac{1}{\sqrt{13}}\approx1-\frac{1}{3.6056}\approx1-0.2774=0.7226.
   $$

4. Update

$$
   x_1=7-\frac{-0.2112}{0.7226}\approx7+0.2924=7.2924.
   $$

---

**Iteration 2:**

1. Compute $$\sqrt{x_1+6}=\sqrt{7.2924+6}=\sqrt{13.2924}\approx3.6459$$.

2. Evaluate

$$
   f(7.2924)=7.2924-2(3.6459)\approx7.2924-7.2918=0.0006.
   $$

3. Compute

$$
   f'(7.2924)=1-\frac{1}{3.6459}\approx1-0.2743=0.7257.
   $$

4. Update

$$
   x_2=7.2924-\frac{0.0006}{0.7257}\approx7.2924-0.0008=7.2916.
   $$

The difference between $$x_2$$ and $$x_1$$ is

$$
\left|7.2924-7.2916\right|\approx0.0008<0.001,
$$

so the iteration stops.

---

**Final Approximations:**

- Newton's Method gives

$$
  x\approx7.2916.
  $$

- For the graphing utility, solving $$x=2\sqrt{x+6}$$ (or graphing $$f(x)=x-2\sqrt{x+6}$$ and finding its zero) yields the same solution. In exact form, squaring both sides gives

$$
  x^2-4x-24=0,
  $$
  
  and the valid (nonnegative) solution is
  
  $$
  x=2+2\sqrt7\approx7.2916.
  $$

---

**Answers:**

Newton's method: $$ x \approx 7.2916 $$  
Graphing utility: $$ x \approx 7.2916 $$
Newton's method: $$ x \approx 7.2916 $$ Graphing utility: $$ x \approx 7.2916 $$

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