Final answer:
To determine if a quadratic equation has two real number solutions, we calculate the discriminant (∑b² - 4ac). Equations 1, 3, and 4 have positive discriminants, therefore each has two real number solutions, while Equations 2 and 5 do not meet the criteria for two real solutions.
Step-by-step explanation:
To determine whether a quadratic equation has two real number solutions, we need to calculate the discriminant, which is the part of the quadratic formula ∑b² - 4ac. For a quadratic equation of the form ax² + bx + c = 0, if the discriminant is positive, the equation has two real and distinct solutions. We substitute the values for a, b, and c from the equations into the discriminant formula for each equation.
- Equation 1: 2x² - 7x - 9, a = 2, b = -7, c = -9. Discriminant = (-7)² - 4(2)(-9) = 49 + 72 = 121.
- Equation 2: x² - 4x + 4, a = 1, b = -4, c = 4. Discriminant = (-4)² - 4(1)(4) = 16 - 16 = 0.
- Equation 3: 4x² - 3x - 1, a = 4, b = -3, c = -1. Discriminant = (-3)² - 4(4)(-1) = 9 + 16 = 25.
- Equation 4: x² - 2x - 8, a = 1, b = -2, c = -8. Discriminant = (-2)² - 4(1)(-8) = 4 + 32 = 36.
- Equation 5: 3x² + 5x + 3, a = 3, b = 5, c = 3. Discriminant = (5)² - 4(3)(3) = 25 - 36 = -11.
Equations with discriminants 1, 3, and 4 have positive values, indicating that they have two real number solutions. Equation 2 has a discriminant of 0, implying one real solution, and Equation 5 has a negative discriminant, which means it has no real solutions.