Question
\( 4\times ^{2}-20\times -24=0 \)
Ask by Colon Dawson. in Mexico
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
Las soluciones son \( x = 6 \) y \( x = -1 \).
Solution
Claro, resolvamos la ecuación cuadrática paso a paso.
Dada la ecuación:
\[ 4x^2 - 20x - 24 = 0 \]
**Paso 1: Simplificar la ecuación**
Primero, podemos simplificar la ecuación dividiendo todos los términos por 4:
\[ x^2 - 5x - 6 = 0 \]
**Paso 2: Aplicar la fórmula cuadrática**
La fórmula cuadrática para resolver \( ax^2 + bx + c = 0 \) es:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
En nuestra ecuación simplificada:
- \( a = 1 \)
- \( b = -5 \)
- \( c = -6 \)
Sustituyendo estos valores en la fórmula:
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \]
\[ x = \frac{5 \pm \sqrt{25 + 24}}{2} \]
\[ x = \frac{5 \pm \sqrt{49}}{2} \]
\[ x = \frac{5 \pm 7}{2} \]
**Paso 3: Calcular las soluciones**
1. Primera solución:
\[ x = \frac{5 + 7}{2} = \frac{12}{2} = 6 \]
2. Segunda solución:
\[ x = \frac{5 - 7}{2} = \frac{-2}{2} = -1 \]
**Conclusión:**
Las soluciones de la ecuación cuadrática \( 4x^2 - 20x - 24 = 0 \) son:
\[ x = 6 \quad \text{y} \quad x = -1 \]
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Simplify this solution Extra Insights
Let's break down the equation step by step! First, to solve \( 4x^2 - 20x - 24 = 0 \), you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 4 \), \( b = -20 \), and \( c = -24 \). After plugging in these values and simplifying, you'll find the roots of this quadratic equation! Remember to check if you can simplify the equation first! In this case, you can divide the whole equation by 4, yielding \( x^2 - 5x - 6 = 0 \). Then, you can either factor it directly or apply the quadratic formula, but factoring might be a fun challenge since it breaks down to \( (x - 6)(x + 1) = 0 \). Now, you have the solutions \( x = 6 \) and \( x = -1 \)! Happy solving!
