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Answer
Las soluciones de la ecuación son \( x = 6 \) y \( x = -8 \).
Solution
Claro, resolvamos la ecuación cuadrática:
\[ x^{2} + 2x - 48 = 0 \]
Para resolver esta ecuación, utilizaremos la **fórmula cuadrática**, que es:
\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
Donde:
- \( a = 1 \)
- \( b = 2 \)
- \( c = -48 \)
Sustituyendo estos valores en la fórmula:
\[ x = \frac{-2 \pm \sqrt{(2)^{2} - 4 \cdot 1 \cdot (-48)}}{2 \cdot 1} \]
\[ x = \frac{-2 \pm \sqrt{4 + 192}}{2} \]
\[ x = \frac{-2 \pm \sqrt{196}}{2} \]
\[ x = \frac{-2 \pm 14}{2} \]
Esto nos da dos soluciones posibles:
1. **Primera solución:**
\[ x = \frac{-2 + 14}{2} \]
\[ x = \frac{12}{2} \]
\[ x = 6 \]
2. **Segunda solución:**
\[ x = \frac{-2 - 14}{2} \]
\[ x = \frac{-16}{2} \]
\[ x = -8 \]
**Por lo tanto, las soluciones de la ecuación son:**
\[ x = 6 \quad \text{y} \quad x = -8 \]
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Simplify this solution Mind Expander
Did you know that the equation \( x^{2}+2 x-48=0 \) can be solved using the quadratic formula? The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is your best friend here, where \( a=1 \), \( b=2 \), and \( c=-48 \). This will give you the roots of the quadratic, showing you where it intersects the x-axis! Now, about common pitfalls: When applying the quadratic formula, some might forget to calculate the discriminant \( b^2 - 4ac \) correctly. A positive discriminant means two distinct real roots, zero means one real root, and a negative discriminant indicates no real solutions at all. Double-check your math to avoid those blunders!
