Bot GPT 4-o mini Let the side length of the smaller square room be \( x \) meters. Consequently, the side length of the larger square room, which is 3 meters longer on each side, will be \( x + 3 \) meters.
The area of the smaller room can be expressed as: \[ \text{Area of the smaller room} = x^2 \]
The area of the larger room can be expressed as: \[ \text{Area of the larger room} = (x + 3)^2 \]
According to the problem, the total area of the two rooms is 89 m²: \[ x^2 + (x + 3)^2 = 89 \]
Expanding the equation: \[ x^2 + (x^2 + 6x + 9) = 89 \]
Combining like terms gives: \[ 2x^2 + 6x + 9 = 89 \]
Subtracting 89 from both sides results in: \[ 2x^2 + 6x + 9 - 89 = 0 \] \[ 2x^2 + 6x - 80 = 0 \]
Next, we can simplify the equation by dividing everything by 2: \[ x^2 + 3x - 40 = 0 \]
Now we will factor the quadratic equation: \[ (x + 8)(x - 5) = 0 \]
This gives us two possible solutions for \( x \): \[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \quad (\text{not a valid solution since dimensions can't be negative}) \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
Thus, the side length of the smaller room is: \[ x = 5 , \text{meters} \]
The side length of the larger room is: \[ x + 3 = 5 + 3 = 8 , \text{meters} \]
In conclusion, the dimensions of the two rooms are:
- Smaller room: \( 5 \) meters x \( 5 \) meters
- Larger room: \( 8 \) meters x \( 8 \) meters
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