To find the possible dimensions of the rectangle, we need to factor the trinomial x² + 3x – 28. The goal is to express this equation in the form of (x + a)(x + b) = 0, where a and b are constants.
First, we look for two numbers that multiply to -28 (the constant term) and add up to 3 (the coefficient of the linear term). The pair of numbers that satisfies these conditions is 7 and -4 because:
- 7 * -4 = -28
- 7 + (-4) = 3
Now we can rewrite the trinomial:
(x + 7)(x - 4) = 0
Next, we set each factor to zero to find the values of x:
x + 7 = 0 => x = -7
x - 4 = 0 => x = 4
Since dimensions cannot be negative, we discard x = -7. Therefore, the possible value for x in this context is 4. This means one possible dimension of the rectangle is 4 units.
To find the other dimension, we can substitute x back into one of our factor equations. If we take the value of x = 4 and plug it into either of the factor equations:
Length (x + 7) = 4 + 7 = 11
Thus, the dimensions of the rectangle can be expressed as:
- Length: 11 units
- Width: 4 units
In conclusion, the dimensions of the rectangular room can be 11 units and 4 units.