Multiple choice

If one root of $x^{2}+ax+8=0$ is $4$ and the equation $x^{2}+ax+b=0$ has equal roots, then $b=$

  1. $7$
  2. $9$
  3. $1$
  4. $3$
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B Correct answer
Explanation

If 4 is a root of x^2 + ax + 8 = 0, then 16 + 4a + 8 = 0, so 4a = -24, a = -6. The second equation is x^2 - 6x + b = 0. For equal roots, the discriminant D = b^2 - 4ac = 0. Here, (-6)^2 - 4(1)(b) = 0, so 36 - 4b = 0, b = 9.

AI explanation

Since x = 4 is a root of x^2 + ax + 8 = 0, substitute it to find the sum of the roots, which means 16 + 4a + 8 = 0, yielding a = -6. The second equation x^2 + ax + b = 0 then becomes x^2 - 6x + b = 0. For this equation to have equal roots, its discriminant must be zero, so (-6)^2 - 4(1)b = 0, which simplifies to 36 - 4b = 0. Solving this gives b = 9.

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