The graph is that of y = 2x2 - 5x - 3. For what value of x will y be negative? For what value of x will y be negative?

Analyze the question: We need the values of x for which y = 2x² - 5x - 3 is negative, meaning the graph lies below the x-axis.

Concept: For a quadratic with a positive leading coefficient (a = 2 > 0), the parabola opens upward. It is negative only between its two roots, and strictly between them, because at the roots y = 0, which is neither negative nor positive.

Solution:

Step 1: Find the roots by factorising 2x² - 5x - 3 = 0.

• We need two numbers that multiply to 2 × (-3) = -6 and add to -5. These are -6 and +1.
• 2x² - 6x + x - 3 = 0
• 2x(x - 3) + 1(x - 3) = 0
• (2x + 1)(x - 3) = 0

So x = -1/2 or x = 3.

Step 2: Since the parabola opens upward, y < 0 between the roots:

2x² - 5x - 3 < 0 ⟹ -1/2 < x < 3

Step 3: Check with a test value, x = 0: y = -3, which is negative, and 0 lies between -1/2 and 3. ✔ Check with x = 4: y = 32 - 20 - 3 = 9, which is positive, so outside the interval y is positive. ✔

At x = -1/2 and x = 3, y = 0, so these endpoints are excluded. This rules out options A, B and D, which include one or both endpoints.

Answer: Option C, -1/2 < x < 3.

Tip: For "y is negative" or "y < 0" questions, the endpoints (the roots) are never included, so use strict inequalities. Include them only when the question says "y ≤ 0". For an upward-opening parabola, negative means between the roots. For a downward-opening one, negative means outside the roots.