Multiple choice

If the roots of the equation a(b - c)x2 + b(c - a)x + c(a - b) = 0 are equal, then which of the following is true?

  1. b = (a + c)/ac

  2. 2/b = (1/a) + (1/c)

  3. 2b = (1/a) + (1/c)

  4. abc = ab + bc + ca
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For equal roots, the discriminant D = b^2 - 4ac = 0. Here, the equation is Ax^2 + Bx + C = 0 where A=a(b-c), B=b(c-a), C=c(a-b). The condition D=0 leads to [b(c-a)]^2 = 4[a(b-c)][c(a-b)]. Simplifying this identity results in 2/b = 1/a + 1/c.

AI explanation

Apply the condition for equal roots, which requires the discriminant to be zero: b^2(c - a)^2 - 4 * a(b - c) * c(a - b) = 0. Factoring this expression yields b^2(c - a)^2 + 4ac(b - c)(a - b) = 0, which can be rewritten as [b(c - a) + 2c(a - b)]^2 = 0. Solving the inner equation gives b(c - a) + 2ac - 2bc = 0, simplifying to 2ac = ab + bc. Dividing the entire equation by abc produces the true statement 2/b = 1/c + 1/a.

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