5x^2 + 10x + 16 = 0 How many distinct real solutions does the given equation have?...

1. TRANSLATE the equation to identify coefficients

• Given equation: \(5x^2 + 10x + 16 = 0\)
• In standard form \(ax^2 + bx + c = 0\):
  • \(a = 5\)
  • \(b = 10\)
  • \(c = 16\)

2. INFER the strategy needed

• To find the number of real solutions for any quadratic equation, we use the discriminant
• The discriminant tells us: if positive (2 solutions), if zero (1 solution), if negative (0 solutions)

3. SIMPLIFY the discriminant calculation

• Discriminant = \(b^2 - 4ac\)
• Substitute our values: \((10)^2 - 4(5)(16)\)
• Calculate: \(100 - 320 = -220\)

4. INFER what the discriminant value means

• Since \(-220 \lt 0\), the discriminant is negative
• Negative discriminant means zero real solutions

Answer: D. Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating \(4(5)(16)\), perhaps getting \(4(5)(16) = 200\) instead of 320, leading to a discriminant of \(100 - 200 = -100\) (still negative) or other miscalculations that could yield a positive result.

If they incorrectly calculate the discriminant as positive, this may lead them to select Choice B (Exactly two).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember what discriminant values mean for the number of solutions. They might calculate the discriminant correctly as -220 but then guess about what a negative value indicates.

This leads to confusion and random guessing among the answer choices.

The Bottom Line:

This problem requires both computational accuracy and conceptual understanding of how discriminants work. Students who know the discriminant formula but make calculation errors, or those who calculate correctly but don't remember the interpretation rules, will struggle to reach the correct answer.