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Rearrange the equation to standard form: r^2 + (q - 2)r + 55 = 0. For no real solutions, the discriminant must be strictly negative: b^2 - 4ac < 0. Thus, (q - 2)^2 - 4(1)(55) < 0, which gives (q - 2)^2 < 220. Taking the square root gives q - 2 < sqrt(220) ≈ 14.832, meaning q < 16.832. The largest integer is 16.

Extracted Problem (Vision OCR)

Desmos Calculator Operations

Analytical & Algebraic Solution

Step 1: Write in Standard Quadratic Form

Alternative Solving Methods

Discriminant Inequality (Algebraic)

Desmos Slider / Direct Testing

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