Common applications and pitfalls

Solving a real-world problem

Quadratic equations can be used to model many situations: the area of a rectangle, the trajectory of a projectile, a company’s profit.

Example: A rectangular garden has a perimeter of 20 m and an area of 24 m². If L and l are the length and width, respectively, then L + l = 10 and L * l = 24. Introducing the unknown x, we obtain x² - 10x + 24 = 0 (using the sum-product property). delta = 100 – 96 = 4, sqrt(4) = 2. x₁ = (10 – 2)/2 = 4, x₂ = (10 + 2)/2 = 6. The dimensions are 4 m and 6 m.

Common pitfalls to avoid

Pitfall	Consequence
Forgetting that a ≠ 0	Confusing it with a first-degree equation
Dividing by x without checking x = 0	Loss of a solution
Forgetting one of the two roots	Incomplete solution
Sign error in δ or in -b	Incorrect roots
Confusing δ < 0 with the complete absence of solutions	In reality, no real solutions only

Checking your solutions

Essential step: substitute each root found back into the original equation to confirm that it does indeed satisfy it. This is the best way to detect a calculation error.

In summary

To solve ax^2+bx+c=0: identify a, b, c -> calculate delta -> apply the formulas according to the sign of delta -> check the solutions. This systematic method works in all cases.