# Linear Quadratic Equation Made Simple: Definitions, Examples, and Real-Life Uses
## 1. What is a “linear quadratic equation”? The phrase describes a **system** of two equations: - One **linear**: y = mx + b - One **quadratic**: y = ax² + bx + c
Solving the system means finding points (x, y) that satisfy **both** equations—graphically the intersections of a straight line and a parabola.
## 2. Quick example Solve: y = 2x + 3 (linear) y = x² – 4x + 7 (quadratic)
Set equal: 2x + 3 = x² – 4x + 7 Rearrange: x² – 6x + 4 = 0 Use quadratic formula: x = 3 ± √5 Corresponding y-values: y = 2(3 ± √5) + 3 = 9 ± 2√5
Intersection points: (3 + √5, 9 + 2√5) and (3 – √5, 9 – 2√5)
## 3. Three solution methods
| Method | When to use | Key steps | |--------|-------------|-----------| | Graphical | Estimate or check | Plot both, read intersections | | Substitution | One equation solved for y | Replace y in quadratic, solve ax²+bx+c=mx+d | | Elimination | Both in standard form | Subtract equations → quadratic = 0 |
## 4. Real-life situations
### 4.1 Projectile & linear path A drone flies straight y = 5x + 10 (height in m) while a ball is thrown along y = –0.1x² + 4x + 5. When do they collide? Solve –0.1x² + 4x + 5 = 5x + 10 → x² + 10x + 50 = 0 → Δ < 0; no collision.
### 4.2 Revenue & cost Revenue R(x) = –2x² + 100x (price x), cost C(x) = 20x + 300. Break-even when R = C: –2x² + 100x = 20x + 300 → –2x² + 80x – 300 = 0 → x = 5 or 30 units.
### 4.3 Optics A parabolic mirror y = 0.25x² intersects a laser beam y = –0.5x + 2. Find focal hits: 0.25x² = –0.5x + 2 → x² + 2x – 8 = 0 → x = 2 or –4 → points (2, 1) and (–4, 4).
## 5. Common pitfalls - Forgetting to back-substitute y - Losing minus signs when rearranging - Confusing “no solution” (Δ < 0, line misses parabola) with “no real roots”
## 6. Fast check list 1. Set equations equal → standard quadratic 2. Compute discriminant Δ 3. Solve for x, then y 4. Interpret context (units, negative values)
## 7. Try these 1. y = x + 4 & y = x² – 2x – 3 2. y = –3x + 5 & y = 2x² + x – 7
Answers: 1) (–1, 3) and (7, 11); 2) (–3, 14) and (2, –1)
## 8. Key takeaway A linear quadratic equation is really a **system**: line + parabola. Solve by setting them equal, solve the resulting quadratic, and always interpret the solutions in context—whether that’s a drone, a factory, or a laser beam.