A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Sets and inequalities
CSCA Mathematics · Lesson 1 of 16Algebra and equations
Algèbre et équations · الجبر والمعادلات
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Most CSCA mathematics questions start with an equation. Solving it cleanly and checking your answer is the most important skill in the paper.
Key ideas
- A linear equation ax + b = c has one solution: x = (c − b) / a.
- A quadratic ax² + bx + c = 0 can be solved by factorising or with the quadratic formula.
- The discriminant Δ = b² − 4ac tells you how many real solutions: Δ > 0 two, Δ = 0 one, Δ < 0 none.
- For a system of two linear equations, use substitution or elimination.
Formulas to know
- x = (−b ± √Δ) / (2a), Δ = b² − 4ac
- Sum of roots = −b/a, product of roots = c/a
- (a + b)² = a² + 2ab + b², a² − b² = (a − b)(a + b)
Worked example
Solve x² − 5x + 6 = 0.
- Look for two numbers with product 6 and sum −5: −2 and −3.
- Factorise: (x − 2)(x − 3) = 0.
- So x − 2 = 0 or x − 3 = 0.
Answer: x = 2 or x = 3
Practice questions
1. Solve 3x + 7 = 22.
x = 3 · x = 5 ✓ · x = 7 · x = 29/3
3x = 15, so x = 5.
2. How many real solutions does x² + 2x + 5 = 0 have?
0 ✓ · 1 · 2 · Infinitely many
Δ = 4 − 20 = −16 < 0, so there is no real solution.
3. Solve the system: x + y = 10 and x − y = 4.
x = 6, y = 4 · x = 7, y = 3 ✓ · x = 4, y = 6 · x = 5, y = 5
Adding the equations: 2x = 14, so x = 7 and y = 3.