If the equations x$^2$ - 5x + 6 = 0 and x$^2$ + px + 6 = 0 have the same roots, find the value of p.

Analyze the question: Two quadratic equations share the same roots, so they must be identical equations. We need the value of p that makes this true.

Concept: For a quadratic x² + bx + c = 0 with roots α and β:

• Sum of roots = -b
• Product of roots = c

Solution:

Step 1: Solve the first equation, x² - 5x + 6 = 0.

Factorise: (x - 2)(x - 3) = 0, so x = 2 or x = 3.

Step 2: The second equation, x² + px + 6 = 0, must have the same roots (2 and 3).

Sum of roots = 2 + 3 = 5, and for the second equation the sum of roots = -p.

So -p = 5, which gives p = -5.

Step 3: Check the product: 2 × 3 = 6, which matches the constant term 6 in the second equation. ✓

Verify by substitution: with p = -5 the second equation becomes x² - 5x + 6 = 0, which is exactly the first equation. ✓

Answer: C (-5)

Tip: When two quadratics with the same leading coefficient have the same roots, they are identical, so just compare the coefficients directly: p = -5. Also remember that the sum of roots is -b/a, so the sign flips. Many students wrongly pick +5 (option A) by forgetting this.