Question

10 Consider the equation (x - p) (x - 6) +1 = 0
having integral coefficients. If the equation has
integral roots, then what values can p have?
(a) 4 or 8 (b) 5 or 10 (c) 6 or 12 (d) 3 or 6​

Answer 1

Answer:

4 or 8

Option A

Explanation:

(x-p)(x-6)+1=0

x^2-(p+6)x+6p+1=0

If this equation has equal roots then discriminant is 0.

Discriminant is

${b}^{2} - 4ac = 0 \\ {(p + 6)}^{2} - 4 \times 1 \times (6p + 1) = 0 \\ {p}^{2} + 36 + 12p - 24p - 4 = 0 \\ {p}^{2} - 12p + 32 = 0 \\ {p}^{2} - 8p - 4p - 32 = 0 \\ p(p - 8) - 4(p - 8) = 0 \\ (p - 8)(p - 4) = 0 \\ p = 8 \: \: or \: 4$

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