Understanding the Quadratic Equation
The quadratic equation x² + 5x + 6 = 0 is in the standard form ax² + bx + c = 0, where a = 1, b = 5, and c = 6. To solve for x using factoring, identify two numbers that multiply to c (6) and add to b (5). These numbers are 2 and 3, since 2 × 3 = 6 and 2 + 3 = 5. Thus, the equation factors as (x + 2)(x + 3) = 0.
Applying the Zero Product Property
Set each factor equal to zero: x + 2 = 0 gives x = -2, and x + 3 = 0 gives x = -3. These are the roots of the equation. Verify by expanding (x + 2)(x + 3) = x² + 5x + 6, which matches the original, confirming the factoring is correct.
Practical Example in Context
Consider a real-world scenario like projectile motion, where the height h(t) = -16t² + 5t + 6 models an object's path (adjusted for units). Setting h(t) = 0 and factoring as (t + 2)(t + 3) = 0 shows the object hits the ground at t = -2 (discard as time can't be negative) and t = -3 (also invalid), but illustrates how factoring finds key times efficiently.
Importance and Common Applications
Factoring quadratics is essential in algebra, physics, and engineering for solving problems involving parabolas, optimization, and motion. It avoids the quadratic formula when possible, simplifying calculations. A common misconception is that all quadratics factor easily over integers; if not, use the formula, but here it works perfectly.