Short Answer 
The quadratic equation x¬¨‚⧠– 34x + c = 0 will have no real solutions when the constant ‘c’ exceeds 289, determined by calculating the discriminant (b¬¨‚⧠– 4ac). The inequality 34¬¨‚⧠– 4c < 0 simplifies to c > 289, indicating that 289 is the least value for ‘c’ for this condition.
Step 1: Understand the Quadratic Equation
The quadratic equation in question is represented as x¬¨‚⧠– 34x + c = 0. Here, ‘c’ is a constant that determines the nature of the roots of the equation. The equation can either have real solutions or no real solutions based on the value of ‘c’. To delve deeper, it is essential to find when this equation has no real solutions.
Step 2: Calculate the Discriminant
The next step involves calculating the discriminant, which is crucial for determining the nature of the roots. The discriminant is given by the formula b¬¨‚⧠– 4ac, where ‘a’ is 1, ‘b’ is -34, and ‘c’ is the constant in the equation. We set up the inequality:
- 34¬≤ Р4(1)(c) < 0
- This can be computed as 1156 – 4c < 0.
Step 3: Solve for the Least Possible Value of n
To find the least possible value of ‘n’, we need to isolate ‘c’ from the inequality we established. Reorganizing the equation gives:
- 4c > 1156
- c > 289
Thus, the least possible value of ‘n’ that ensures there are no real solutions for the quadratic equation is 289.