To solve the problem step by step, we will analyze the given equations and find the required value of \( \sin^2 A + \sin^2 B + \sin^2 C \).
### Step 1: Identify the equations
We have two quadratic equations:
1. \( ax^2 + 2bx + 3c = 0 \)
2. \( 3x^2 + 8x + 15 = 0 \)
### Step 2: Determine the nature of the roots of the second equation
To find the nature of the roots of the second equation, we calculate the discriminant \( D \):
\[
D = b^2 - 4ac = 8^2 - 4 \cdot 3 \cdot 15 = 64 - 180 = -116
\]
Since \( D < 0 \), the roots of the second equation are imaginary.
### Step 3: Analyze the common root condition
Since the first equation has real coefficients and the second equation has imaginary roots, the only way they can have a common root is if the common root is also imaginary.
### Step 4: Set up the ratio of coefficients
For both equations to have a common root, the ratios of the coefficients must be equal:
\[
\frac{a}{3} = \frac{2b}{8} = \frac{3c}{15}
\]
This simplifies to:
\[
\frac{a}{3} = \frac{b}{4} = \frac{c}{5}
\]
Let \( a = 3k \), \( b = 4k \), and \( c = 5k \) for some \( k \).
### Step 5: Check if \( a, b, c \) form a triangle
We need to verify if \( a, b, c \) satisfy the triangle inequality:
\[
(3k)^2 + (4k)^2 = (5k)^2
\]
Calculating this gives:
\[
9k^2 + 16k^2 = 25k^2
\]
This holds true, confirming that \( a, b, c \) can represent the sides of a right triangle.
### Step 6: Find the sine values
In a right triangle with sides \( 3k, 4k, \) and \( 5k \):
- The angle opposite to side \( 3k \) is \( A \).
- The angle opposite to side \( 4k \) is \( B \).
- The angle opposite to side \( 5k \) is \( C \) (the right angle).
Using the definitions of sine:
\[
\sin A = \frac{3k}{5k} = \frac{3}{5}, \quad \sin B = \frac{4k}{5k} = \frac{4}{5}, \quad \sin C = 1
\]
### Step 7: Calculate \( \sin^2 A + \sin^2 B + \sin^2 C \)
Now we compute:
\[
\sin^2 A = \left(\frac{3}{5}\right)^2 = \frac{9}{25}
\]
\[
\sin^2 B = \left(\frac{4}{5}\right)^2 = \frac{16}{25}
\]
\[
\sin^2 C = 1^2 = 1 = \frac{25}{25}
\]
Adding these together:
\[
\sin^2 A + \sin^2 B + \sin^2 C = \frac{9}{25} + \frac{16}{25} + \frac{25}{25} = \frac{50}{25} = 2
\]
### Final Answer
Thus, the value of \( \sin^2 A + \sin^2 B + \sin^2 C \) is \( 2 \).
