Multiple choice

If the roots of the equation x2 – bx + c = 0 differ by 2, then which of the following is true?

  1. c2 = 4(c + 1)

  2. b2 = 4c + 4

  3. c2 = b + 4

  4. b2 = 4(c + 2)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let roots be r and r+2. Sum = 2r+2 = b, so r = (b-2)/2. Product = r(r+2) = c. Substitute r: ((b-2)/2) * ((b+2)/2) = c. (b^2 - 4) / 4 = c. b^2 - 4 = 4c. b^2 = 4c + 4.

AI explanation

Let the roots be alpha and beta, so their difference alpha minus beta equals 2. Squaring both sides gives alpha squared plus beta squared minus 2 times alpha times beta equals 4. Using the identities alpha plus beta equals b and alpha times beta equals c, we substitute to get (alpha plus beta) squared minus 4 times alpha times beta equals 4. Replacing these with b and c gives b squared minus 4c equals 4, which rearranges to b squared equals 4c plus 4.

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