Bot GPT 4-o mini To convert the number \( 142_r \) from base \( r \) to base 10, we need to evaluate it using the formula for base conversion. The number in base \( r \) can be expressed as:
\[ 1 \cdot r^2 + 4 \cdot r^1 + 2 \cdot r^0 \]
This simplifies to:
\[ 1 \cdot r^2 + 4 \cdot r + 2 \]
According to the problem, this is equal to \( 47_{10} \). Therefore, we have:
\[ r^2 + 4r + 2 = 47 \]
Now, subtract 47 from both sides:
\[ r^2 + 4r + 2 - 47 = 0 \]
This simplifies to:
\[ r^2 + 4r - 45 = 0 \]
Next, we can use the quadratic formula to solve for \( r \):
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 4 \), and \( c = -45 \). Substituting these values into the formula gives:
\[ r = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-45)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 4^2 - 4 \cdot 1 \cdot (-45) = 16 + 180 = 196 \]
Taking the square root,
\[ \sqrt{196} = 14 \]
Now substituting back into the formula for \( r \):
\[ r = \frac{-4 \pm 14}{2} \]
This results in two possible solutions:
- \( r = \frac{10}{2} = 5 \)
- \( r = \frac{-18}{2} = -9 \) (not valid for a base)
Thus, the only valid solution is:
\[ r = 5 \]
Hence, the base \( r \) is \( 5 \).
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