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Answer
Las soluciones son \( x = 6 \) y \( x = -8 \).
Solution
Para resolver la ecuación cuadrática \( x^{2} + 2x - 48 = 0 \), podemos utilizar la **fórmula general**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
**Paso 1: Identificar los coeficientes**
En la ecuación dada:
- \( a = 1 \)
- \( b = 2 \)
- \( c = -48 \)
**Paso 2: Calcular el discriminante (\( \Delta \))**
El discriminante se calcula como:
\[
\Delta = b^2 - 4ac
\]
Sustituyendo los valores:
\[
\Delta = (2)^2 - 4(1)(-48) = 4 + 192 = 196
\]
**Paso 3: Calcular las raíces utilizando la fórmula general**
\[
x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-2 \pm \sqrt{196}}{2 \cdot 1} = \frac{-2 \pm 14}{2}
\]
Esto nos da dos soluciones:
1. **Primera solución:**
\[
x = \frac{-2 + 14}{2} = \frac{12}{2} = 6
\]
2. **Segunda solución:**
\[
x = \frac{-2 - 14}{2} = \frac{-16}{2} = -8
\]
**Conclusión:**
Las soluciones de la ecuación \( x^{2} + 2x - 48 = 0 \) son:
\[
x = 6 \quad \text{y} \quad x = -8
\]
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Simplify this solution Bonus Knowledge
To solve the equation \( x^{2}+2x-48=0 \), you can factor it as \( (x + 8)(x - 6) = 0 \). This gives the solutions \( x = -8 \) and \( x = 6 \). Easy peasy! Now, let’s delve into the significance of quadratic equations! These equations, like the one you've just seen, are prevalent in various fields like physics for projectile motion, engineering for designing structures, and finance for modeling profit and costs. Quadratics are everywhere, helping us understand the relationships between variables in dynamic systems!
