Question

1. lawson is pouring a rectangular concrete slab for his doghouse. the slab will cover an area of $5x^2 + 12x - 9$ square feet. write expressions that represent possible dimensions for the length and width of the concrete slab.

Explanation:

Step1: Factor the quadratic expression

We need to factor the quadratic expression \(5x^2 + 12x - 9\). We look for two numbers that multiply to \(5\times(-9)= -45\) and add up to \(12\). The numbers are \(15\) and \(-3\) since \(15\times(-3)= -45\) and \(15 + (-3)=12\).
So we rewrite the middle term:
\(5x^2 + 15x - 3x - 9\)
Now we group the terms:
\((5x^2 + 15x)+(-3x - 9)\)
Factor out the greatest common factor from each group:
\(5x(x + 3)-3(x + 3)\)
Now we can factor out \((x + 3)\):
\((5x - 3)(x + 3)\)

Step2: Identify the length and width

Since the area of a rectangle is length times width, and we have factored the area expression \(5x^2 + 12x - 9\) into \((5x - 3)(x + 3)\), the possible expressions for the length and width are \(5x - 3\) and \(x + 3\) (or vice versa, since length and width are interchangeable in terms of multiplication for area).

Answer:

The possible expressions for the length and width of the concrete slab are \(\boldsymbol{5x - 3}\) and \(\boldsymbol{x + 3}\) (or \(x + 3\) and \(5x - 3\)).