60-second revision
- For ax² + bx + c = 0: sum of roots = −b/a, product = c/a.
- Discriminant D = b² − 4ac: D > 0 unequal real roots, D = 0 equal, D < 0 no real roots.
- Split b into two numbers whose product is ac; the roots are those numbers ÷ a with signs flipped.
- Both signs (b, c) positive: both roots negative. b negative, c positive: both positive.
- c negative: one root positive and one negative.
- x² = 16 gives 4 or −4, but sqrt(16) = 4 only.
- Equation with roots p and q: x² − (p + q)x + pq = 0.
The basics
A quadratic is ax² + bx + c = 0 with a not zero. Its two roots are [−b + sqrt(b² − 4ac)] ÷ 2a and [−b − sqrt(b² − 4ac)] ÷ 2a. In bank exams you rarely need the formula: the equations are built to factorise.
| Discriminant D = b² − 4ac | Nature of roots |
|---|---|
| D > 0 and a perfect square | Real, unequal, rational |
| D > 0, not a perfect square | Real, unequal, irrational |
| D = 0 | Real and equal (each = −b/2a) |
| D < 0 | No real roots |
Factorise in seconds
Find two numbers whose product is a × c and whose sum is b. The roots are those two numbers divided by a, with their signs flipped.
Worked example: 2x² − 11x + 15 = 0. a × c = 30; −5 and −6 multiply to 30 and add to −11. Flip signs and divide by 2: roots 5/2 and 6/2, i.e. 2.5 and 3.
Worked example: 3x² − 10x + 8 = 0. a × c = 24; split −10 as −4 and −6. Roots 4/3 and 6/3 = 2. Check: 3 × 4 − 20 + 8 = 0.
Comparing x and y (bank format)
Solve both equations, list the roots, and compare every x with every y. There are at most four pairs.
| What you find | Answer |
|---|---|
| Every x bigger than every y | x > y |
| Every x bigger or equal, at least one pair equal | x >= y |
| Every x smaller than every y | x < y |
| Every x smaller or equal, at least one pair equal | x <= y |
| Both have the same single value | x = y |
| Some pairs one way, some the other | Relationship cannot be established |
Worked example: I. x² − 9x + 20 = 0 gives 4 and 5. II. y² − 7y + 12 = 0 gives 3 and 4. Pairs: 4 vs 3 (>), 4 vs 4 (=), 5 vs 3 (>), 5 vs 4 (>). Answer: x >= y.
Worked example: I. x² = 49 gives 7 or −7. II. y = sqrt(49) = 7 only. Pairs: 7 = 7, −7 < 7. Answer: x <= y.
Sum and product of roots
- p + q = −b/a and pq = c/a.
- p² + q² = (p + q)² − 2pq.
- 1/p + 1/q = (p + q)/pq.
- (p − q)² = (p + q)² − 4pq.
- Equation from roots p and q: x² − (p + q)x + pq = 0.
Worked example: for 2x² − 8x + 5 = 0, the sum is 4 and the product is 5/2, so 1/p + 1/q = 4 ÷ 5/2 = 8/5.
Worked example: the equation whose roots are 4 and −7 is x² − (−3)x + (−28) = 0, i.e. x² + 3x − 28 = 0.
Unknown coefficients
Equal roots: set D = 0. For x² + kx + 9 = 0, k² = 36, so k = 6 or −6.
One root given: substitute it. If 3 is a root of x² − 8x + k = 0, then 9 − 24 + k = 0, so k = 15 and the other root is 8 − 3 = 5.
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