Multiple choice

If the quadratic equation $ ax^{2}+2 cx+b=0 $ and $ ax^{2}+2 bx+c=0 (b\neq c) $ have a common root, then $a+4b+4c=$

  1. $-2$
  2. $-1$
  3. $0$
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Subtracting the two equations: a(x^2-x^2) + 2c(x) - 2b(x) + b - c = 0. 2x(c-b) - (c-b) = 0. Since b != c, we can divide by (c-b) to get 2x - 1 = 0, so x = 1/2. Substituting x=1/2 into the first equation: a(1/4) + 2c(1/2) + b = 0. a/4 + c + b = 0. Multiplying by 4 gives a + 4c + 4b = 0.

AI explanation

For the two quadratic equations ax^2 + 2cx + b = 0 and ax^2 + 2bx + c = 0 to have a common root, let that root be r. Equating the two expressions for ar^2 gives -2cr - b = -2br - c, which simplifies to r = (c - b) / (2(c - b)) = 1/2. Substituting r = 1/2 back into the first equation yields a(1/2)^2 + 2c(1/2) + b = 0, which reduces to a + 4c + 4b = 0. Therefore, the value of a + 4b + 4c is 0.

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