Quadratic equations
A quadratic equation has an x² term. Solve it by factorising, or with the quadratic formula when the factors are hard to find.
What the discriminant tells you
| D = b² − 4ac | Roots | Example |
|---|---|---|
| D > 0 and a perfect square | Two rational roots; it factorises | x² − 5x − 14 = 0: D = 81 |
| D > 0, not a perfect square | Two irrational roots; use the formula | 2x² − 3x − 4 = 0: D = 41 |
| D = 0 | One repeated root | x² − 6x + 9 = 0: x = 3 |
| D < 0 | No real roots | x² + 2x + 5 = 0: D = −16 |
Solving a quadratic
Solve by factorising
Solve x² − 5x − 14 = 0.
- Find two numbers with product −14 and sum −5: −7 and +2.
- So (x − 7)(x + 2) = 0.
- Either x − 7 = 0, giving x = 7, or x + 2 = 0, giving x = −2.
- Check x = 7: 49 − 35 − 14 = 0. ✓ Check x = −2: 4 + 10 − 14 = 0. ✓
x = 7 or x = −2
Solve with the formula
Solve 2x² − 3x − 4 = 0, giving answers to 2 decimal places.
- a = 2, b = −3, c = −4.
- D = b² − 4ac = (−3)² − 4(2)(−4) = 9 + 32 = 41.
- x = [3 ± √41] ÷ 4, and √41 = 6.4031.
- x = (3 + 6.4031) ÷ 4 = 9.4031 ÷ 4 = 2.35.
- x = (3 − 6.4031) ÷ 4 = −3.4031 ÷ 4 = −0.85.
x = 2.35 or x = −0.85
Two break-even points
A small Negombo boatyard's monthly profit (Rs. '000) is P = −x² + 120x − 2,000, where x is the number of repair jobs. (a) Find the break-even outputs. (b) Over what range is there a profit? (c) Find the profit at 60 jobs.
- (a) Break-even means P = 0: −x² + 120x − 2,000 = 0.
- Multiply by −1: x² − 120x + 2,000 = 0.
- Two numbers with product 2,000 and sum −120: −20 and −100. So (x − 20)(x − 100) = 0.
- x = 20 or x = 100.
- (b) The parabola opens downward (a = −1), so profit is positive between the roots: 20 < x < 100.
- (c) P(60) = −3,600 + 7,200 − 2,000 = 1,600, i.e. Rs. 1,600,000.
Break-even at 20 and 100 jobs; profitable between them; profit at 60 jobs is Rs. 1,600,000.
When a Sri Lankan firm's profit rises and then falls as output grows (overtime pay, waste, price cuts), profit is a quadratic. Its two roots are the two break-even outputs; profit is made only between them.
Short answer: “How many real roots does x² + 2x + 5 = 0 have? Why?” None: D = 4 − 20 = −16 < 0.
Structured question: “Solve 3x² − 5x − 2 = 0 by factorising and check with the formula.” (3x + 1)(x − 2) = 0, x = 2 or −1/3. Formula: D = 49, x = (5 ± 7)/6.
Essay: “Profit P = −x² + 120x − 2,000. Find the break-even outputs and the profit at x = 60.” x² − 120x + 2,000 = 0 gives x = 20, 100; P(60) = 1,600.
"Minus b, plus or minus the root of b squared minus 4ac, all over 2a." Say it aloud until it sticks. Then look at D first: it tells you whether factorising will work.