QUESTION 1 – FULL SOLUTIONS

1.1 Solve for x:

1.1.1 Watch

x² – 2x – 24 = 0
Factor the equation:
(x – 6)(x + 4) = 0
Answers: x = 6 or x = -4

1.1.2
2x² – 3x – 3 = 0
Use the quadratic formula:
x = [3 ± √(9 + 24)] / 4
x = (3 ± √33) / 4
√33 ≈ 5.7446
Answers: x ≈ 2.19 or x ≈ -0.69 (to 2 decimal places)

1.1.3 Wacth
x² + 5x ≤ -4
x² + 5x + 4 ≤ 0
(x + 1)(x + 4) ≤ 0
Test intervals:
This inequality is true when -4 ≤ x ≤ -1
Answer: x ∈ [-4, -1]

1.1.4 Watch 
√(x + 28) = 2 – x
Square both sides:
x + 28 = (2 – x)²
x + 28 = 4 – 4x + x²
Simplify:
x² – 5x – 24 = 0
(x – 8)(x + 3) = 0
Check both values:
x = 8 → √(36) = 6, but 2 – 8 = -6 → Not valid
x = -3 → √(25) = 5 and 2 – (-3) = 5 → Valid
Answer: x = -3

1.2 Solve simultaneously for x and y: Watch

Given:
2y = 3 + x → x = 2y – 3
2xy + 7 = x² + 4y²
Substitute x:
2(2y – 3)y + 7 = (2y – 3)² + 4y²
4y² – 6y + 7 = 4y² – 12y + 9 + 4y²
Simplify:
0 = 4y² – 12y + 9 + 4y² – 4y² + 6y – 7
0 = 4y² – 6y + 2
Use the quadratic formula:
y = [6 ± √(36 – 32)] / 8
y = [6 ± √4] / 8
y = (6 ± 2) / 8
y = 1 or 0.5
Now find x:
If y = 1, x = 2(1) – 3 = -1
If y = 0.5, x = 2(0.5) – 3 = -2
Answers:
(x, y) = (-1, 1) or (-2, 0.5)

1.3 Prove that x is a non-real number: Watch

The root of the equation is:
x = [-n ± √(n² – 4mp)] / 2m
Given that m, n, p are positive real numbers in a geometric sequence.
In a geometric sequence

NSC Mathematics P1 November 2021 – Question 1 Explained

This article provides a full step-by-step explanation of Question 1 from the NSC Mathematics Paper 1, November 2021. Perfect for learners, tutors, and education platforms targeting high CPC keywords like solve quadratic equations, simultaneous equations, algebra help, and non-real solutions.

✅ QUESTION 1: Solve and Understand Algebra Step-by-Step

1.1 Solve for x:

1.1.1   x² - 2x - 24 = 0

This is a quadratic equation. Factor the expression:

Find two numbers that multiply to -24 and add to -2:

(x - 6)(x + 4) = 0

So:

• x = 6
• x = -4

Final Answer: x = 6 or x = -4

1.1.2   2x² - 3x - 3 = 0 (Correct to TWO decimal places)

Use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Where a = 2, b = -3, c = -3:

x = (3 ± √(9 + 24)) / 4 = (3 ± √33) / 4

√33 ≈ 5.74, so:

• x ≈ (3 + 5.74)/4 = 2.19
• x ≈ (3 – 5.74)/4 = -0.69

Final Answer: x ≈ 2.19 or x ≈ -0.69

1.1.3   x² + 5x ≤ -4

Move all terms to one side:

x² + 5x + 4 ≤ 0

Factor it:

(x + 1)(x + 4) ≤ 0

Use a number line:

• Critical points: x = -4 and x = -1
• Test intervals show the inequality is true for x ∈ [-4, -1]

Final Answer: x ∈ [-4, -1]

1.1.4   √(x + 28) = 2 - x

Square both sides:

x