If the equations px² + 3qx + r = 0 and px² + 3rx + q = 0 have a common root, p + 9q + 9r is equal to:

By cross-multiplying the coefficients of the two equations (a1, b1, c1 and a2, b2, c2), we use the condition (c1*a2 - c2*a1)^2 = (a1*b2 - a2*b1)(b1*c2 - b2*c1) to find the common root. Substituting the given values yields (rp - rp)^2 = (3pr - 3pr)(3q^2 - 3r^2), which is not directly helpful, so instead subtract the two equations to get 3x(q - r) + (r - q) = 0. Factoring out (q - r) gives 3x - 1 = 0, so the common root is x = 1/3. Substituting x = 1/3 into the first equation results in p/9 + q + r = 0, and multiplying the entire expression by 9 gives p + 9q + 9r = 0.

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