QUESTION 4 4.1 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) 2 Given: $$ p=\sqrt{\mathrm{b}^{2}+4 \mathrm{ac}} $$ 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 $$ 0,6 \dot{3} $$ 4.3.2 $$ 0,12 \dot{4} $$

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Sure, let's work through each part of **QUESTION 4** step by step.

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### **4.1 Round off the following numbers to the decimal places indicated:**

#### **4.1.1**  
**Number:** 4.3768534  
**Decimal Places to Round:** 4

**Solution:**  
To round off to 4 decimal places, we look at the fifth decimal digit to decide whether to round up or stay the same.

- **Original Number:** 4.3768534  
- **Fourth Decimal Place Digit:** 8  
- **Fifth Decimal Place Digit:** 5

Since the fifth digit is **5**, we round up the fourth digit by 1.

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

---

#### **4.1.2**  
**Number:** $$ \sqrt{7.24357} $$  
**Decimal Places to Round:** 1

**Solution:**  
First, calculate the square root of 7.24357.

- $$ \sqrt{7.24357} \approx 2.6927 $$

To round to one decimal place, we look at the second decimal digit:

- **First Decimal Place Digit:** 6  
- **Second Decimal Place Digit:** 9

Since the second digit is **9** (which is ≥5), we round up the first decimal digit.

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

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### **4.2 Given:**  
$$ p = \sqrt{b^2 + 4ac} $$

#### **4.2.1 Determine the value of $$ p $$ if $$ a = 2; b = 6; c = 2 $$**  
**Solution:**  
Plug the given values into the formula:

$$
p = \sqrt{b^2 + 4ac} = \sqrt{6^2 + 4 \times 2 \times 2} = \sqrt{36 + 16} = \sqrt{52}
$$

Simplify $$ \sqrt{52} $$:

$$
\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}
$$

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

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#### **4.2.2 State whether $$ p $$ is rational or irrational.**  
**Solution:**  
- $$ \sqrt{13} $$ is an **irrational** number.
- Multiplying an irrational number by a rational number (2) still results in an **irrational** number.

**Answer:**  
$$ p $$ is **irrational**.

---

#### **4.2.3 Between which TWO consecutive integers does $$ p $$ lie?**  
**Solution:**  
Calculate the approximate value of $$ p $$:

$$
\sqrt{13} \approx 3.6055 \
p = 2 \times 3.6055 \approx 7.211
$$

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

---

### **4.3 Write the following recurring decimals in fraction form:**

#### **4.3.1**  
**Recurring Decimal:** $$ 0.\overline{6}3 $$ (Assuming the notation $$ 0,6\dot{3} $$ represents 0.6333...)

**Solution:**  
Let $$ x = 0.6333\ldots $$

$$
10x = 6.3333\ldots \
x = 0.6333\ldots \
\text{Subtracting: } 10x - x = 6.3333\ldots - 0.6333\ldots \
9x = 5.7 \
x = \frac{5.7}{9} = \frac{57}{90} = \frac{19}{30}
$$

**Answer:**  
$$ 0.\overline{6}3 = \frac{19}{30} $$

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#### **4.3.2**  
**Recurring Decimal:** $$ 0.\overline{12}4 $$ (Assuming the notation $$ 0,12\dot{4} $$ represents 0.12444...)

**Solution:**  
Let $$ x = 0.12444\ldots $$

$$
1000x = 124.4444\ldots \
100x = 12.4444\ldots \
\text{Subtracting: } 1000x - 100x = 124.4444\ldots - 12.4444\ldots \
900x = 112 \
x = \frac{112}{900} = \frac{28}{225}
$$

**Answer:**  
$$ 0.\overline{12}4 = \frac{28}{225} $$

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I hope this helps! Let me know if you have any further questions.
**Simplified Answers:** 1. **4.1.1** **Rounded Number:** 4.3769 2. **4.1.2** **Rounded Value:** 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** $$ 0.\overline{6}3 = \frac{19}{30} $$ 7. **4.3.2** $$ 0.\overline{12}4 = \frac{28}{225} $$

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