One fourth of a herd of goats was seen in the forest. Twice the square root of the number in the herd had gone up the hill and the remaining 15 goats were on the bank of the river. Find the total number of goats.

Answer

To solve the problem step by step, let's denote the total number of goats in the herd as \( X \).

### Step 1: Set up the equation based on the problem statement.
According to the problem:
- One fourth of the herd was seen in the forest: \( \frac{1}{4}X \)
- Twice the square root of the number in the herd had gone up the hill: \( 2\sqrt{X} \)
- The remaining goats were on the bank of the river: 15 goats

From this information, we can set up the equation:
\[ \frac{1}{4}X + 2\sqrt{X} + 15 = X \]

### Step 2: Rearrange the equation.
To simplify the equation, we can move all terms to one side:
\[ \frac{1}{4}X + 2\sqrt{X} + 15 - X = 0 \]
This simplifies to:
\[ -\frac{3}{4}X + 2\sqrt{X} + 15 = 0 \]
Multiplying through by 4 to eliminate the fraction gives:
\[ -3X + 8\sqrt{X} + 60 = 0 \]

### Step 3: Rearranging further.
Rearranging the equation gives us:
\[ 3X - 8\sqrt{X} - 60 = 0 \]

### Step 4: Substitute \( \sqrt{X} \) with \( y \).
Let \( y = \sqrt{X} \). Then \( X = y^2 \). Substituting this into the equation gives:
\[ 3y^2 - 8y - 60 = 0 \]

### Step 5: Solve the quadratic equation.
Now we can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3, b = -8, c = -60 \):
\[ b^2 - 4ac = (-8)^2 - 4 \cdot 3 \cdot (-60) = 64 + 720 = 784 \]
Now substituting back into the quadratic formula:
\[ y = \frac{8 \pm \sqrt{784}}{6} = \frac{8 \pm 28}{6} \]
Calculating the two possible values for \( y \):
1. \( y = \frac{36}{6} = 6 \)
2. \( y = \frac{-20}{6} = -\frac{10}{3} \) (ignore this as \( y \) must be non-negative)

### Step 6: Find \( X \).
Since \( y = \sqrt{X} \), we have:
\[ \sqrt{X} = 6 \implies X = 6^2 = 36 \]

### Conclusion:
The total number of goats in the herd is \( \boxed{36} \).

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