Multiple choice

For what values of $k$, the equation $x^2-2(1+3k)x+7(3+2k)=0$ has equal roots ?

  1. $1 $ and $-\dfrac {10}{9}$
  2. $2$ and $-\dfrac {10}{9}$
  3. $3$ and $-\dfrac {10}{9}$
  4. $4$ and $-\dfrac {10}{9}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For equal roots, discriminant D = b^2 - 4ac = 0. Here, 4(1+3k)^2 - 4(7)(3+2k) = 0. (1+9k^2+6k) - (21+14k) = 0. 9k^2 - 8k - 20 = 0. Solving for k: (9k+10)(k-2) = 0. k = 2 or -10/9.

AI explanation

A quadratic equation has equal roots when its discriminant is zero, meaning b^2 - 4ac = 0. Substituting the coefficients gives [-2(1+3k)]^2 - 4(1)(7)(3+2k) = 0. Simplifying this yields 4(1+6k+9k^2) - 84 - 56k = 0, which further reduces to the quadratic equation 9k^2 - 8k - 20 = 0. Factoring this as (9k+10)(k-2) = 0 gives the final result of 2 and -10/9.

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