Answer:
Step-by-step explanation:
1. **Completing the Table**:
| Expresión Algebraica | Ecuación Cuadrática (y=0) | Discriminante (D) | Soluciones de la Ecuación | Intersecciones con el eje x |
|----------------------|--------------------------|--------------------|--------------------------|-----------------------------|
| y = 3x² + 2x - 8 | 3x² + 2x - 8 = 0 | \( D = 2^2 - 4*3*(-8) = 100 \) | \( x = \frac{-2 \pm \sqrt{100}}{6} = \frac{-2 \pm 10}{6} \) | \( x = -2 \) and \( x = \frac{4}{3} \) |
| y = 4x² + 6x + 4 | 4x² + 6x + 4 = 0 | \( D = 6^2 - 4*4*4 = 4 \) | \( x = \frac{-6 \pm \sqrt{4}}{8} = \frac{-6 \pm 2}{8} \) | \( x = -1 \) |
| y = x² + 6x + 8 | x² + 6x + 8 = 0 | \( D = 6^2 - 4*1*8 = 4 \) | \( x = \frac{-6 \pm \sqrt{4}}{2} = \frac{-6 \pm 2}{2} \) | \( x = -4 \) and \( x = -2 \) |
| y = -x² - 2x - 48 | -x² - 2x - 48 = 0 | \( D = (-2)^2 - 4*(-1)*(-48) = 208 \) | \( x = \frac{2 \pm \sqrt{208}}{-2} \) | \( x \) is complex (no real solutions) |
| y = 2x² - 10x + 9 | 2x² - 10x + 9 = 0 | \( D = (-10)^2 - 4*2*9 = 4 \) | \( x = \frac{10 \pm \sqrt{4}}{4} = \frac{10 \pm 2}{4} \) | \( x = 3 \) and \( x = \frac{1}{2} \) |
2. **Interpretation**:
- The table shows the quadratic expressions, their corresponding quadratic equations with \( y = 0 \), the calculated discriminants, solutions of the equations, and the x-intersections with the x-axis for each given expression.
- Depending on the discriminant value, we can determine the nature of the solutions (real, repeated, or complex) and the number of x-intercepts each equation has.
By filling in and analyzing the table, we can understand the relationship between the given algebraic expressions, quadratic equations, discriminants, solutions, and their intersections with the x-axis
