Question 1209641
Here's how to find the values of *a* for which the given quadratic equation has only one distinct root:

1. **Rewrite the equation in standard form:**

   Combine like terms to get the equation in the form Ax² + Bx + C = 0:

   (a - 5)x² + (7 - 23)x + (2 + 12) = 0
   (a - 5)x² - 16x + 14 = 0

2. **Apply the discriminant condition:**

   A quadratic equation has only one distinct root (a double root) when its discriminant (B² - 4AC) is equal to zero.  In our equation:

   * A = (a - 5)
   * B = -16
   * C = 14

   So, we set the discriminant equal to zero:

   (-16)² - 4 * (a - 5) * 14 = 0
   256 - 56(a - 5) = 0
   256 - 56a + 280 = 0
   536 - 56a = 0

3. **Solve for *a*:**

   56a = 536
   a = 536 / 56
   a = 67/7

Therefore, the only nonzero constant *a* for which the given equation has only one distinct root is a = 67/7.