Let the width of the courtyard be \( w \) yards. According to the problem, the length \( l \) of the courtyard can be expressed in terms of the width as follows:
\[ l = 2w - 3 \]
The area \( A \) of a rectangle is given by the formula:
\[ A = l \times w \]
We are given that the area is 104 square yards:
\[ lw = 104 \]
Substituting the expression for \( l \) into the area equation:
\[ (2w - 3)w = 104 \]
Expanding this equation:
\[ 2w^2 - 3w = 104 \]
Rearranging it into standard quadratic form gives:
\[ 2w^2 - 3w - 104 = 0 \]
Now we can use the quadratic formula to find \( w \). The quadratic formula is given by:
\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( 2w^2 - 3w - 104 = 0 \), we have:
- \( a = 2 \)
- \( b = -3 \)
- \( c = -104 \)
Computing the discriminant \( b^2 - 4ac \):
\[ (-3)^2 - 4 \times 2 \times (-104) = 9 + 832 = 841 \]
Taking the square root of the discriminant:
\[ \sqrt{841} = 29 \]
Now applying the quadratic formula:
\[ w = \frac{-(-3) \pm 29}{2 \times 2} = \frac{3 \pm 29}{4} \]
Calculating the two possible values for \( w \):
- \( w = \frac{3 + 29}{4} = \frac{32}{4} = 8 \)
- \( w = \frac{3 - 29}{4} = \frac{-26}{4} = -6.5 \) (not valid since width cannot be negative)
Thus, we have:
\[ w = 8 \text{ yards} \]
Now we can find the length \( l \):
\[ l = 2w - 3 = 2(8) - 3 = 16 - 3 = 13 \text{ yards} \]
Therefore, the length of the playground is:
\[ \boxed{13} \text{ yards} \]