Question
Real Tutor Solution
Tutor-Verified Answer
Answer
Las soluciones son \( x = 2 \) y \( x = 7 \).
Solution
Para resolver la ecuación cuadrática \( x^{2} - 9x + 14 = 0 \), podemos utilizar el método de factorización. A continuación, se detallan los pasos:
1. **Identificar los coeficientes:**
- \( a = 1 \) (coeficiente de \( x^{2} \))
- \( b = -9 \) (coeficiente de \( x \))
- \( c = 14 \) (término constante)
2. **Buscar dos números que multiplicados den \( a \cdot c = 1 \cdot 14 = 14 \) y que sumados den \( b = -9 \):**
- Los números son **-7** y **-2**, ya que:
\[
(-7) \times (-2) = 14 \quad \text{y} \quad (-7) + (-2) = -9
\]
3. **Factorizar la ecuación utilizando estos números:**
\[
x^{2} - 7x - 2x + 14 = 0
\]
\[
x(x - 7) - 2(x - 7) = 0
\]
\[
(x - 2)(x - 7) = 0
\]
4. **Aplicar el principio del producto cero:**
\[
x - 2 = 0 \quad \text{o} \quad x - 7 = 0
\]
5. **Resolver para \( x \):**
\[
x = 2 \quad \text{o} \quad x = 7
\]
**Conclusión:** Las soluciones de la ecuación \( x^{2} - 9x + 14 = 0 \) son \( x = 2 \) y \( x = 7 \).
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Simplify this solution Beyond the Answer
To solve the quadratic equation \( x^{2}-9 x+14=0 \), we can factor it as \( (x-7)(x-2)=0 \). This gives us the solutions \( x = 7 \) and \( x = 2 \). It's a classic "factor and solve" scenario that many students encounter. When working with quadratic equations, one common mistake is forgetting to set both factors equal to zero after factoring. Always remember that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). Double-checking your factorization and ensuring both potential solutions are considered will save you from missing answers!
