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Largest q for No Real Solutions
r2 + qr = 8r − 87
In the given equation, q is an integer constant. The given equation has no real solutions. What is the largest possible value of q?
Need a hint?
Move all terms to one side and use the discriminant condition for no real solutions.
Correct answer
26
Solution
Rewrite the equation:
r2 + qr = 8r − 87
r2 + (q − 8)r + 87 = 0
For no real solutions, the discriminant must be negative:
(q − 8)2 − 4(1)(87) < 0
(q − 8)2 < 348
Since √348 is between 18 and 19, the greatest integer value of q occurs when:
q − 8 = 18
q = 26
Detailed solution
The discriminant must be strictly less than zero. Since (q−8)2 must be less than 348, the greatest integer value for q−8 is 18, giving q=26.
Common mistake
A common mistake is rounding √348 up to 19 and choosing 27. But 192 = 361, which is too large.
Trap logic
The trap is using ≤ or rounding the square root incorrectly.
What this reveals
If this was difficult, it may show a weak spot in discriminant inequalities.
StudyGlitch notes
This SAT advanced algebra question tests quadratic discriminants and integer constraints.
For no-real-solution questions, use discriminant less than zero.
Since 182 = 324 and 192 = 361, use 18.
This week’s question is more than a score.
A wrong answer usually points to a pattern: slow recognition, a trap choice, weak setup, or pressure under time. Use this challenge as a signal, then connect it to your diagnostic, practice history, and weekly StudyGlitch progress.