Algebra | Criticalyx Mathematics

🎯 Learning Objectives🎯 Objektif Pembelajaran

Simplify and evaluate algebraic expressionsPermudahkan dan nilaikan ungkapan algebra

Solve linear equations and inequalitiesSelesaikan persamaan dan ketaksamaan linear

Factorise expressions using common methodsFaktorkan ungkapan menggunakan kaedah biasa

Solve quadratic equations using the quadratic formulaSelesaikan persamaan kuadratik menggunakan formula kuadratik

Identify and apply direct, inverse and joint variationKenal pasti dan aplikasi ubahan terus, songsang dan gabungan

📖 Key Concepts📖 Konsep Utama

Algebraic Expressions (Form 1)Ungkapan Algebra (Tingkatan 1)

Terms separated by + or −. Like terms can be combined.Sebutan dipisahkan oleh + atau −. Sebutan serupa boleh digabungkan.

3x + 5x = 8x (like terms) | 3x + 5y (unlike — cannot combine)

Linear Equations (Form 1)Persamaan Linear (Tingkatan 1)

Variables with power 1 only. Solve by isolating the variable.Pemboleh ubah dengan kuasa 1 sahaja. Selesaikan dengan mengasingkan pemboleh ubah.

ax + b = c → x = (c − b) / a

Factorisation (Form 2)Pemfaktoran (Tingkatan 2)

Reversing expansion — writing as a product of factors.Membalikkan pengembangan — menulis sebagai hasil darab faktor.

xy + xz = x(y + z) | x² − y² = (x+y)(x−y) | x² + bx + c = (x+p)(x+q)

Quadratic Equations (Form 4)Persamaan Kuadratik (Tingkatan 4)

Equations where the highest power is 2. Use the quadratic formula when factorisation is difficult.Persamaan di mana kuasa tertinggi ialah 2. Guna formula kuadratik apabila pemfaktoran sukar.

x = (−b ± √(b²−4ac)) / 2a
Discriminant: b²−4ac > 0 (2 roots) | = 0 (1 root) | < 0 (no real roots)

Variation (Form 5)Ubahan (Tingkatan 5)

Direct, inverse and joint variation. Always find k first!Ubahan terus, songsang dan gabungan. Sentiasa cari k dahulu!

Direct: y = kx | Inverse: y = k/x | Joint: y = kx/z

✏️ Worked Examples✏️ Contoh Penyelesaian

Solve 3(2x − 4) = 5x + 2Selesaikan 3(2x − 4) = 5x + 2

Step 1: Expand brackets — 6x − 12 = 5x + 2Langkah 1: Kembangkan kurungan — 6x − 12 = 5x + 2

Step 2: Collect x terms — 6x − 5x = 2 + 12Langkah 2: Kumpul sebutan x — 6x − 5x = 2 + 12

x = 14

Solve 2x² − 7x + 3 = 0 using the quadratic formulaSelesaikan 2x² − 7x + 3 = 0 menggunakan formula kuadratik

a = 2, b = −7, c = 3

b² − 4ac = 49 − 24 = 25

x = (7 ± √25) / 4 = (7 ± 5) / 4

x = 12/4 = 3 or x = 2/4 = ½

x = 3 or x = ½

y varies directly as x. When x = 4, y = 12. Find y when x = 7.y berubah terus dengan x. Bila x = 4, y = 12. Cari y bila x = 7.

y = kx → 12 = k(4) → k = 3

y = 3(7) = 21

y = 21

Factorise x² + 5x + 6Faktorkan x² + 5x + 6

Find p, q where p + q = 5 and pq = 6 → p = 2, q = 3Cari p, q di mana p + q = 5 dan pq = 6 → p = 2, q = 3

(x + 2)(x + 3)

Solve the inequality 3x − 5 > 7Selesaikan ketaksamaan 3x − 5 > 7

3x > 7 + 5

3x > 12

x > 4

⚠️ Common Mistakes⚠️ Kesilapan Biasa

❌ Sign errors expanding bracketsRalat tanda mengembangkan kurungan

−3(x − 2) = −3x + 6 (NOT −3x − 6). The minus flips ALL signs inside the brackets.−3(x − 2) = −3x + 6 (BUKAN −3x − 6). Tanda tolak songsangkan SEMUA tanda di dalam kurungan.

❌ Forgetting to flip inequality signLupa songsangkan tanda ketaksamaan

When multiplying/dividing by a negative number: −2x > 6 → x < −3 (flip > to <)Apabila darab/bahagi dengan nombor negatif: −2x > 6 → x < −3 (songsangkan > jadi <)

❌ Not checking quadratic solutionsTidak memeriksa penyelesaian kuadratik

Always substitute back to verify. If x = 3 solves x² − 5x + 6 = 0: 9 − 15 + 6 = 0 ✓Sentiasa gantikan balik untuk mengesahkan. Jika x = 3 selesaikan x² − 5x + 6 = 0: 9 − 15 + 6 = 0 ✓

❌ Confusing variation typesMengelirukan jenis ubahan

Direct: y = kx (both increase). Inverse: y = k/x (one up, one down). Read the question carefully!Terus: y = kx (kedua meningkat). Songsang: y = k/x (satu naik, satu turun). Baca soalan dengan teliti!

📝 SPM Practice📝 Amalan SPM

Paper 1 — Multiple ChoiceKertas 1 — Pilihan Berganda

Simplify 4(2x − 3) − 2(x + 1)Permudahkan 4(2x − 3) − 2(x + 1)

6x − 14

6x − 10

10x − 14

10x − 10

Solve 3x − 5 > 7Selesaikan 3x − 5 > 7

x > 4

x > 2

x < 4

x > 12

Factorise x² − 9 completelyFaktorkan x² − 9 secara lengkap

(x − 3)²

(x + 3)(x − 3)

(x − 9)(x + 1)

x(x − 9)

y varies directly as x². When x = 2, y = 20. Find y when x = 3.y berubah terus dengan x². Bila x = 2, y = 20. Cari y bila x = 3.

The solutions of x² − 5x + 6 = 0 arePenyelesaian x² − 5x + 6 = 0 ialah

x = −2, x = −3

x = 2, x = 3

x = −2, x = 3

x = 1, x = 6

Paper 2 — Structured QuestionsKertas 2 — Soalan Berstruktur

6 marks6 markah

(a) Solve 2x² + 3x − 2 = 0. [4 marks]
(b) Find k if one root of 2x² + 3x − 2 = kx is ½. [2 marks](a) Selesaikan 2x² + 3x − 2 = 0. [4 markah]
(b) Cari k jika satu punca 2x² + 3x − 2 = kx ialah ½. [2 markah]

Solution (a)Penyelesaian (a)

a = 2, b = 3, c = −2

b² − 4ac = 9 + 16 = 25

x = (−3 ± √25) / 4 = (−3 ± 5) / 4

x = 2/4 = ½ or x = −8/4 = −2

x = ½ or x = −2 [4 marks4 markah]

Solution (b)Penyelesaian (b)

Substitute x = ½: 2(¼) + 3(½) − 2 = k(½)Gantikan x = ½: 2(¼) + 3(½) − 2 = k(½)

½ + 3/2 − 2 = k/2 → 0 = k/2

k = 0 [2 marks2 markah]

5 marks5 markah

y varies inversely as x. When x = 3, y = 4.
(a) Express y in terms of x. [2 marks]
(b) Find y when x = 6. [3 marks]y berubah songsang dengan x. Bila x = 3, y = 4.
(a) Ungkapkan y dalam sebutan x. [2 markah]
(b) Cari y bila x = 6. [3 markah]

Solution (a)Penyelesaian (a)

y = k/x → 4 = k/3 → k = 12

y = 12/x [2 marks2 markah]

Solution (b)Penyelesaian (b)

y = 12/6 = 2

y = 2 [3 marks3 markah]

📋 Formula Summary📋 Ringkasan Formula

Algebra Key Formulas:Formula Utama Algebra:

Quadratic Formula:Formula Kuadratik: x = (−b ± √(b²−4ac)) / 2a

Factorisation:Pemfaktoran:
xy + xz = x(y + z) | x² − y² = (x+y)(x−y) | x² + bx + c = (x+p)(x+q)

Variation:Ubahan:
Direct:Terus: y = kx | Inverse:Songsang: y = k/x | Joint:Gabungan: y = kx/z

Linear Inequality:Ketaksamaan Linear: Flip sign when × or ÷ by negativeSongsangkan tanda bila × atau ÷ dengan negatif