A curve is such that when y = 0, x = -2 or x = 3. Find the equation of the curve.

Analyze the question: The curve crosses the x-axis (y = 0) at x = -2 and x = 3. These are the roots of the quadratic equation, and we need the equation that has them.

Concept: If a quadratic has roots α and β, then its equation is y = (x - α)(x - β)

Solution:

Roots: x = -2 and x = 3

This gives the factors (x + 2) and (x - 3):

y = (x + 2)(x - 3)

Expand: y = x² - 3x + 2x - 6 y = x² - x - 6

Check with the options:

• Sum of roots = -2 + 3 = 1, so the coefficient of x is -1 ✔
• Product of roots = (-2)(3) = -6, so the constant is -6 ✔

Verify: when x = 3, y = 9 - 3 - 6 = 0 ✔. When x = -2, y = 4 + 2 - 6 = 0 ✔

Tip: For a quadratic with roots α and β, use x² - (sum of roots)x + (product of roots). Here that is x² - (1)x + (-6). Be careful with signs: the factor for x = -2 is (x + 2), not (x - 2).