Equations for CA Foundation
Linear, simultaneous, quadratic and simple cubic equations, with the shortcuts that save time in Paper 3.
Quantitative AptitudePaper 3, Business Mathematics13 MCQs2 lessons2 flashcards3 memory tricks
What you'll learn
- Linear and simultaneous equations
Solve linear equations and pairs of equations quickly. - Quadratic and cubic equations
Find roots and use sum and product of roots.
Key formulas
| Roots | x = [−b ± √(b² − 4ac)] ÷ 2a |
|---|---|
| Sum of roots | α + β = −b/a |
| Product of roots | αβ = c/a |
| Equation from roots | x² − (α + β)x + αβ = 0 |
| Useful identity | α² + β² = (α + β)² − 2αβ |
Concepts explained
Sum and product of roots
For ax² + bx + c = 0 with roots α and β: α + β = −b/a and αβ = c/a.
What it means
- Equation from roots: x² − (α + β)x + αβ = 0.
- α² + β² = (α + β)² − 2αβ.
- (α − β)² = (α + β)² − 4αβ.
- 2x² − 8x + 3 = 0
- Sum = −(−8)/2; product = 3/2
Answer: Sum = 4, product = 1.5
- Roots 2 and −3
- Sum = −1, product = −6
Answer: x² + x − 6 = 0
Related terms
- Factorisation: split the middle term using product and sum
Exam tip: In MCQs, checking sum and product against each option is faster than solving.
Discriminant and nature of roots
The discriminant D = b² − 4ac decides whether the roots are real and different, real and equal, or not real.
What it means
- D > 0: real and unequal (rational if D is a perfect square).
- D = 0: real and equal (x = −b/2a).
- D < 0: not real (imaginary).
- x² − 6x + 9 = 0
- D = 36 − 36 = 0
Answer: Equal roots: x = 3
- x² + x + 1 = 0
- D = 1 − 4 = −3
Answer: Roots not real
Related terms
- Quadratic formula: x = [−b ± √D] ÷ 2a
Exam tip: 'Equal roots' in a question means set b² = 4ac and solve for the unknown.
Simultaneous linear equations
Two equations in two unknowns can be solved by elimination or substitution.
What it means
- Elimination: add or subtract the equations to remove one unknown.
- Substitution: express one unknown from one equation and put it in the other.
- In MCQs, substitute the options back: often the fastest method.
- x + y = 10 and x − y = 4
- Add: 2x = 14
Answer: x = 7, y = 3
Related terms
- Consistent system: has at least one solution
Exam tip: Always check your answer in both equations before marking.
Memory tricks
For ax² + bx + c = 0: α + β = −b/a, αβ = c/a.
b² − 4ac decides whether roots are real and distinct, equal, or not real.
If roots are α and β, the equation is x² − (α + β)x + αβ = 0.
Try 3 CA Foundation-style questions
Q1. If x + y = 10 and x − y = 4, then (x, y) is
- (3, 7)
- (6, 4)
- (5, 5)
- (7, 3)
Show answer
Answer: (4) (7, 3)
Adding: 2x = 14, x = 7; then y = 3.
Q2. If 2 is a root of x² − kx + 8 = 0, then k is
- 4
- −6
- 6
- 8
Show answer
Answer: (3) 6
Substitute x = 2: 4 − 2k + 8 = 0, so 2k = 12 and k = 6.
Q3. The roots of x² − 5x + 6 = 0 are
- 2 and 3
- −2 and −3
- 1 and 6
- −1 and 6
Show answer
Answer: (1) 2 and 3
x² − 5x + 6 = (x − 2)(x − 3), so x = 2 or 3.
Practise all 13 questions Free · explanations for every option · see where you go wrong