Final answer:
To find the values of a for different numbers of solutions for the quadratic equation, we examine the discriminant from the quadratic formula. If the discriminant is zero, there is one real solution; if it is positive, there are two real solutions, and if negative, there are two complex solutions.
Step-by-step explanation:
To determine the number of solutions for the given quadratic equation ax² − 5x + 3 = 0, we must consider the discriminant, which is ∑ = b² - 4ac. The discriminant is part of the quadratic formula, x = −b ± √(b² - 4ac) / (2a), which finds the solutions to a quadratic equation.
a. One real solution: This occurs when the discriminant is zero (∑ = 0). For the equation ax² − 5x + 3 = 0, it means solving (-5)² - 4 * a * 3 = 0.
b. Two real solutions: This occurs when the discriminant is positive (∑ > 0). We need to find values of a for which (-5)² - 4 * a * 3 is greater than zero.
c. Two complex solutions: This occurs when the discriminant is negative (∑ < 0), so we need to find values of a where (-5)² - 4 * a * 3 is less than zero.