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To solve the problem, we start with the equation given:
\[ x^2 - 6x = 16 \]
First, rearrange all terms to one side to form a quadratic equation:
\[ x^2 - 6x - 16 = 0 \]
Next, we will solve this quadratic equation using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -6 \), and \( c = -16 \).
Calculate the discriminant:
\[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-16) \]
\[ = 36 + 64 \]
\[ = 100 \]
Now, find the square root of the discriminant:
\[ \sqrt{100} = 10 \]
Substitute back into the quadratic formula:
\[ x = \frac{-(-6) \pm 10}{2 \cdot 1} \]
\[ x = \frac{6 \pm 10}{2} \]
This gives us two possible solutions for \( x \):
\[ x = \frac{16}{2} = 8 \]
\[ x = \frac{-4}{2} = -2 \]
Since age cannot be negative, we discard \( x = -2 \).
Therefore, the boy's age is \( x = 8 \).
To verify, substitute \( x = 8 \) back into the original context of the problem:
- Square of the boy's age: \( 8^2 = 64 \)
- Six times the girl's age: \( 6 \cdot 8 = 48 \)
- The difference: \( 64 - 48 = 16 \)
The calculations match the conditions given in the problem statement. Thus, the boy's age is confirmed to be {8}