QUESTION 4 Round off the following numbers to the decimal places indicated: 4.1.1 4,3768534 ( 4 decimal places $$ ) $$ 4.1.2 $$ \sqrt{7,24357}(1 $$ decimal place $$ ) $$ Given: $$ p=\sqrt{b^{2}+4 \mathrm{ac}} $$ 4.2.1 Determine the value of $$ p $$ if $$ a=2 $$; $$ b=6 $$; $$ c=2 $$ (Leave your answer in the simplest surd form) 4.2.2 State whether $$ p $$ is rational or irrational. 4.2.3 Between which TWO consecutive integers does $$ p $$ lie? Write the following recurring decimals in fraction form: 4.3.1 $$ \begin{array}{l} ext { 4.3 } \ ext { 4. }\end{array} $$

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QUESTION 4 Round off the following numbers to the decimal places indicated: 4.1.1 4,3768534 ( 4 decimal places $$ ) $$ 4.1.2 $$ \sqrt{7,24357}(1 $$ decimal place $$ ) $$ Given: $$ p=\sqrt{b^{2}+4 \mathrm{ac}} $$ 4.2.1 Determine the value of $$ p $$ if $$ a=2 $$; $$ b=6 $$; $$ c=2 $$ (Leave your answer in the simplest surd form) 4.2.2 State whether $$ p $$ is rational or irrational. 4.2.3 Between which TWO consecutive integers does $$ p $$ lie? Write the following recurring decimals in fraction form: 4.3.1 $$ \begin{array}{l}	ext { 4.3 } \ 	ext { 4. }\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's address each part of **QUESTION 4** step by step.

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### **4.1.1 Rounding 4.3768534 to 4 Decimal Places**

**Number:** 4.3768534  
**Decimal Places to Round:** 4

**Steps:**
1. Identify the fourth decimal place: **4.3768**.
2. Look at the fifth decimal place to decide whether to round up or stay: The fifth digit is **5**.
3. Since the fifth digit is **5** or more, round up the fourth decimal place by 1.

**Rounded Number:**  
**4.3769**

---

### **4.1.2 Rounding √7.24357 to 1 Decimal Place**

**Expression:** √7.24357  
**Decimal Places to Round:** 1

**Steps:**
1. Calculate the square root of 7.24357.  
   - √7 ≈ 2.6458  
   - √7.24357 ≈ **2.692** (approximate value).
2. Round to 1 decimal place:
   - The first decimal place is **6**.
   - The second decimal place is **9** (since it's ≥5, round up).

**Rounded Value:**  
**2.7**

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### **4.2.1 Determining the Value of $$ p = \sqrt{b^2 + 4ac} $$ with $$ a = 2 $$, $$ b = 6 $$, $$ c = 2 $$**

**Given:**  
$$ a = 2 $$  
$$ b = 6 $$  
$$ c = 2 $$

**Calculation:**
$$
p = \sqrt{b^2 + 4ac} = \sqrt{6^2 + 4 \times 2 \times 2} = \sqrt{36 + 16} = \sqrt{52}
$$
Simplify $$ \sqrt{52} $$:
$$
\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}
$$

**Value of $$ p $$:**  
$$ p = 2\sqrt{13} $$

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### **4.2.2 Determining Whether $$ p $$ is Rational or Irrational**

**Expression:**  
$$ p = 2\sqrt{13} $$

**Analysis:**
- $$ \sqrt{13} $$ is an **irrational** number because 13 is not a perfect square.
- Multiplying an irrational number by a rational number (2) still results in an **irrational** number.

**Conclusion:**  
$$ p $$ is **Irrational**.

---

### **4.2.3 Identifying the Two Consecutive Integers Between Which $$ p $$ Lies**

**Expression:**  
$$ p = 2\sqrt{13} $$

**Steps:**
1. Approximate $$ \sqrt{13} $$:
   - $$ \sqrt{9} = 3 $$
   - $$ \sqrt{16} = 4 $$
   - So, $$ \sqrt{13} $$ is between **3** and **4**.
   - More precisely, $$ \sqrt{13} \approx 3.6055 $$.
2. Calculate $$ p $$:
   $$
   p = 2 \times 3.6055 \approx 7.211
   $$
3. Identify the consecutive integers:
   - **7** and **8**.

**Conclusion:**  
$$ p $$ lies between **7** and **8**.

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### **4.3.1 Writing Recurring Decimals in Fraction Form**

*It appears there might be some formatting issues with the recurring decimals you intended to provide in part 4.3.1. However, I can demonstrate how to convert a recurring decimal to a fraction using an example.*

**Example:** Convert $$ 0.\overline{6} $$ to a fraction.

**Steps:**
1. Let $$ x = 0.\overline{6} $$.
2. Multiply both sides by 10 to shift the decimal point:
   $$
   10x = 6.\overline{6}
   $$
3. Subtract the original equation from this new equation:
   $$
   10x - x = 6.\overline{6} - 0.\overline{6}
   $$
   $$
   9x = 6
   $$
4. Solve for $$ x $$:
   $$
   x = \frac{6}{9} = \frac{2}{3}
   $$

**Conclusion:**  
$$ 0.\overline{6} = \frac{2}{3} $$

*If you provide the specific recurring decimals for parts 4.3.1 and 4.3.2, I can offer precise conversions.*

---

Feel free to provide the specific recurring decimals for **4.3.1** and **4.3.2**, and I'll be happy to assist further!
**Simplified Answers for QUESTION 4:** 1. **4.1.1** Rounded to 4 decimal places: **4.3769** 2. **4.1.2** Rounded to 1 decimal place: **2.7** 3. **4.2.1** $$ p = 2\sqrt{13} $$ 4. **4.2.2** $$ p $$ is **Irrational** 5. **4.2.3** $$ p $$ lies between **7** and **8** 6. **4.3.1** Example: $$ 0.\overline{6} = \frac{2}{3} $$ (Please provide specific decimals for complete solutions.)

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