Question

If a, b, c, d are distinct integer such that (x − a) (x − b) (x − c) (x − d) = 4 has an integer solution x = r. Find the ratio ​

Answer 1

Given :- If a, b, c and d are distinct integers such that (x-a)(x-b)(x-c)(x-d) = 4 has an integer solution x = r.

To Find :-

• The ratio of (a+b+c+d)/r = ?

Solution :-

→ (x - a) * (x - b) * (x - c) * (x - d) = 4

since a,b,c and d distinct integers ,

→ (x - a) * (x - b) * (x - c) * (x - d) = 1 * (-1) * 2 * (-2) [ note here we can write 1 * (-1) * (-2) * 2 or 2 * 1 * (-2) * (-1) , all have same values .]

comparing we get,

→ x - a = 1 => a = (x - 1)

→ x - b = (-1) => b = (x + 1)

→ x - c = 2 => c = x - 2

→ x - d = (-2) => d = x + 2 .

then,

→ (a + b + c + d)/r

→ (x - 1 + x + 1 + x - 2 + x + 2)/r

→ 4x/r

given that x = r,

therefore,

→ 4r/r

→ 4 .

Hence, the ratio of (a+b+c+d)/r will be (4/1) or we can say that (a + b + c + d) : r = 4 : 1 .

Learn more :-

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if a²+ab+b²=25

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