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Answer
Las soluciones son \( x = -5 \) y \( x = 3 \).
Solution
Para resolver la ecuación cuadrática \( x^{2} + 2x - 15 = 0 \), podemos utilizar el método de factorización o la fórmula cuadrática. A continuación, se presenta la solución utilizando la factorización:
**Paso 1: Identificar dos números que multiplicados den \(-15\) y que sumados den \(2\).**
Buscamos dos números que cumplan con estas condiciones:
- **Multiplicación:** \(5 \times (-3) = -15\)
- **Suma:** \(5 + (-3) = 2\)
Por lo tanto, los números son \(5\) y \(-3\).
**Paso 2: Expresar la ecuación en su forma factorizada.**
\[
x^{2} + 2x - 15 = 0 \\
x^{2} + 5x - 3x - 15 = 0 \\
(x^{2} + 5x) + (-3x - 15) = 0 \\
x(x + 5) - 3(x + 5) = 0 \\
(x + 5)(x - 3) = 0
\]
**Paso 3: Encontrar las soluciones de la ecuación factorizada.**
\[
x + 5 = 0 \quad \Rightarrow \quad x = -5 \\
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
**Conclusión:**
Las soluciones de la ecuación \( x^{2} + 2x - 15 = 0 \) son:
\[
x = -5 \quad \text{y} \quad x = 3
\]
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Simplify this solution Mind Expander
This equation is a classic example of a quadratic equation which can be solved using the factoring method, completing the square, or using the quadratic formula! For this particular equation, it factors nicely into \( (x + 5)(x - 3) = 0 \). This shows that the solutions are \( x = -5 \) and \( x = 3 \). Quadratic equations like this show up all the time in real-world situations, such as calculating projectile trajectories or optimizing area. For example, if you're designing a garden in rectangular shape, the dimensions could be represented by such equations when maximizing space or dealing with certain restrictions—math really does blossom!
