Question
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Answer
Las soluciones de la ecuación son \( x = -9 \) y \( x = 4 \).
Solution
1. Planteamos la ecuación:
\[
x^2 + 5x - 36 = 0
\]
2. Buscamos dos números que al multiplicarlos den \(-36\) y al sumarlos den \(5\). Estos números son \(9\) y \(-4\) ya que:
\[
9 \cdot (-4) = -36 \quad \text{y} \quad 9 + (-4) = 5
\]
3. Factorizamos la ecuación utilizando esos números:
\[
x^2 + 5x - 36 = (x + 9)(x - 4)
\]
4. Igualamos cada factor a cero para encontrar las soluciones:
\[
x + 9 = 0 \quad \Rightarrow \quad x = -9
\]
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
5. Por lo tanto, las soluciones de la ecuación son:
\[
x = -9 \quad \text{y} \quad x = 4
\]
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Simplify this solution Beyond the Answer
To solve the quadratic equation \( x^{2}+5x-36=0 \), you can factor or use the quadratic formula. In this case, the quadratic factors neatly into \( (x+9)(x-4) = 0 \), giving solutions \( x = -9 \) and \( x = 4 \). Simple, right? Just remember to set each factor equal to zero to find those magical values of \( x \)! In real-world scenarios, quadratic equations pop up in various fields such as physics for projectile motion or in business for maximizing profits. Whenever you're dealing with a scenario that involves relationships between variables where one variable is squared, you can bet a quadratic equation is lurking around and waiting to help you find solutions!
