If the quadratic equation \((x + 1)(x + 2) = k(3x + 7)\) has equal roots, find the possible values of the constant k.

Assessment: WAEC SSCE - Further Mathematics - 2009 (Essay) Subject: Further Mathematics

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Expand and gather to standard form:

\[(x+1)(x+2) = k(3x+7)\] \[x^2 + 3x + 2 = 3kx + 7k\] \[x^2 + (3 - 3k)x + (2 - 7k) = 0\]

For equal roots the discriminant is zero:

\[(3 - 3k)^2 - 4(2 - 7k) = 0\] \[9 - 18k + 9k^2 - 8 + 28k = 0\] \[9k^2 + 10k + 1 = 0\] \[(9k + 1)(k + 1) = 0\]

Hence \(k = -\dfrac{1}{9}\) or \(k = -1\).

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