x 2 − 34x + c = 0 In the given equation, c is a constant. The equation has no real solutions if c n. What is the least possible value of n?

Question

x 2 − 34x + c = 0 In the given equation, c is a constant. The equation has no real solutions if c > n. What is the least possible value of n?

Accepted Answer

The correct answer is 289. A quadratic equation of the form ax 2 + bx + c = 0, where a, b, and c are constants, has no real solutions when the value of the discriminant, b 2 − 4ac, is less than 0. In the given equation, x 2 − 34x + c = 0, a = 1 and b = −34. Therefore, the discriminant of the given equation can be expressed as (−34) 2 − 4(1)(c), or 1,156 − 4c. It follows that the given equation has no real solutions when 1,156 − 4c 289. It’s given that the equation x 2 − 34x + c = 0 has no real solutions when c > n. Therefore, the least possible value of n is 289.

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