What is the true statement about the expression 4x^2 + 19x + 5?

The expression 4x² + 19x + 5 is a quadratic equation in the standard form ax² + bx + c, where:

• a = 4
• b = 19
• c = 5

To determine the properties of this quadratic equation, we can analyze its coefficients:

• 1. Parabola Orientation: Since the coefficient of x² (which is 4) is positive, this means that the parabola opens upwards.
• 2. Roots of the Equation: To find out if there are real roots, we can calculate the discriminant, which is given by the formula D = b² – 4ac. In this case:
  • Substituting the values, we have: D = 19² – 4 * 4 * 5
  • This simplifies to: D = 361 – 80 = 281

  Since the discriminant is positive (D > 0), the quadratic equation has two distinct real roots.

• 3. Vertex of the Parabola: The x-coordinate of the vertex can be calculated using the formula x = -b / (2a). For this equation, it would be:
  • Substituting the values:
    x = -19 / (2 * 4) = -19 / 8
  • This gives the vertex x-coordinate at approximately -2.375.

In summary, the true statement regarding 4x² + 19x + 5 is that it is an upward-opening parabola with two distinct real roots.