If `24 sqrt (3)x^(3) + 5 sqrt (5)y^(3) = (2 sqrt(3)x + sqrt(5)y) xx (Ax^(2) - Bxy + Cy^(2)),` then what is the value of `(A^(2) - B^(2) + C^(2)) ?`

Options

A. 189
B. 111
C. 109
D. 169

Answer

To solve the equation \( 24 \sqrt{3} x^3 + 5 \sqrt{5} y^3 = (2 \sqrt{3} x + \sqrt{5} y)(Ax^2 - Bxy + Cy^2) \), we will first expand the right-hand side and then compare coefficients.

### Step 1: Expand the right-hand side
We start with the expression:
\[ (2 \sqrt{3} x + \sqrt{5} y)(Ax^2 - Bxy + Cy^2) \]
Using the distributive property (FOIL method), we expand this:
\[ = 2 \sqrt{3} x \cdot Ax^2 + 2 \sqrt{3} x \cdot (-Bxy) + 2 \sqrt{3} x \cdot Cy^2 + \sqrt{5} y \cdot Ax^2 + \sqrt{5} y \cdot (-Bxy) + \sqrt{5} y \cdot Cy^2 \]
This simplifies to:
\[ = 2A\sqrt{3} x^3 - 2B\sqrt{3} x^2y + 2C\sqrt{3} xy^2 + A\sqrt{5} x^2y - B\sqrt{5} xy^2 + C\sqrt{5} y^3 \]

### Step 2: Combine like terms
Now, we combine the terms:
\[ = (2A\sqrt{3}) x^3 + (A\sqrt{5} - 2B\sqrt{3}) x^2y + (2C\sqrt{3} - B\sqrt{5}) xy^2 + (C\sqrt{5}) y^3 \]

### Step 3: Set coefficients equal
Now we set the coefficients of like terms from both sides of the equation:
1. Coefficient of \(x^3\): \(2A\sqrt{3} = 24\sqrt{3}\)
2. Coefficient of \(y^3\): \(C\sqrt{5} = 5\sqrt{5}\)
3. Coefficient of \(x^2y\): \(A\sqrt{5} - 2B\sqrt{3} = 0\)
4. Coefficient of \(xy^2\): \(2C\sqrt{3} - B\sqrt{5} = 0\)

### Step 4: Solve for \(A\), \(B\), and \(C\)
From the first equation:
\[ 2A\sqrt{3} = 24\sqrt{3} \implies 2A = 24 \implies A = 12 \]

From the second equation:
\[ C\sqrt{5} = 5\sqrt{5} \implies C = 5 \]

From the third equation:
\[ 12\sqrt{5} - 2B\sqrt{3} = 0 \implies 2B\sqrt{3} = 12\sqrt{5} \implies B = \frac{12\sqrt{5}}{2\sqrt{3}} = 6\sqrt{\frac{5}{3}} = 6\frac{\sqrt{15}}{3} = 2\sqrt{15} \]

From the fourth equation:
\[ 2C\sqrt{3} - B\sqrt{5} = 0 \implies 2(5)\sqrt{3} - B\sqrt{5} = 0 \implies 10\sqrt{3} = B\sqrt{5} \implies B = \frac{10\sqrt{3}}{\sqrt{5}} = 10\sqrt{\frac{3}{5}} = 10\frac{\sqrt{15}}{5} = 2\sqrt{15} \]

### Step 5: Calculate \(A^2 - B^2 + C^2\)
Now we have:
- \(A = 12\)
- \(B = 2\sqrt{15}\)
- \(C = 5\)

We calculate:
\[ A^2 = 12^2 = 144 \]
\[ B^2 = (2\sqrt{15})^2 = 4 \cdot 15 = 60 \]
\[ C^2 = 5^2 = 25 \]

Now substituting these values into the expression:
\[ A^2 - B^2 + C^2 = 144 - 60 + 25 = 109 \]

### Final Answer
Thus, the value of \(A^2 - B^2 + C^2\) is \(109\).