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Answer
Las soluciones de la ecuación \( x^{2} - 11x + 24 = 0 \) son \( x = 3 \) y \( x = 8 \).
Solution
Para resolver la ecuación cuadrática \( x^{2} - 11x + 24 = 0 \), podemos seguir los siguientes pasos:
1. **Factorización de la ecuación:**
Buscamos dos números que multiplicados den 24 (el término constante) y sumados den -11 (el coeficiente de \( x \)). Esos números son -3 y -8, ya que:
\[
(-3) \times (-8) = 24 \quad \text{y} \quad (-3) + (-8) = -11
\]
Por lo tanto, la ecuación se puede factorizar de la siguiente manera:
\[
(x - 3)(x - 8) = 0
\]
2. **Aplicar la propiedad del producto nulo:**
Si el producto de dos factores es cero, entonces al menos uno de los factores debe ser cero. Así que:
\[
x - 3 = 0 \quad \text{o} \quad x - 8 = 0
\]
3. **Resolver para \( x \):**
\[
x = 3 \quad \text{y} \quad x = 8
\]
Por lo tanto, las soluciones de la ecuación son:
\[
x = 3 \quad \text{y} \quad x = 8
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
Did you know that the quadratic equation you’re working on has a historical significance? Quadratic equations date back to ancient Babylon, around 2000 BC! The Babylonians had methods to solve them geometrically, showing just how long we've been fascinated by these mathematical puzzles. Now, let's get practical! To solve this equation, you can factor it into \( (x - 3)(x - 8) = 0 \). Setting each factor to zero gives you the solutions \( x = 3 \) and \( x = 8 \). Just remember, always check your factors; many folks overlook negative signs, making simple mistakes while factoring!
