Multiple choice

If the equation $2{ x }^{ 2 }+(a+3)x+8=0$ has equal roots, then one of the values of $a$ is

  1. $-9$
  2. $-5$
  3. $-11$
  4. $11$
  5. $9$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For equal roots, the discriminant D = b^2 - 4ac must be 0. Here, (a+3)^2 - 4(2)(8) = 0, so (a+3)^2 - 64 = 0. Thus (a+3)^2 = 64, meaning a+3 = 8 or a+3 = -8. This gives a = 5 or a = -11.

AI explanation

For a quadratic equation ax^2 + bx + c = 0 to have equal roots, the discriminant (b^2 - 4ac) must equal zero. Substituting the coefficients from 2x^2 + (a+3)x + 8 = 0 gives (a+3)^2 - 4(2)(8) = 0. Expanding this yields a^2 + 6a + 9 - 64 = 0, which simplifies to a^2 + 6a - 55 = 0. Factoring the quadratic equation gives (a + 11)(a - 5) = 0, so the values of a are -11 and 5.

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