The width of the rectangle is $4$ feet and the length is $6$ feet.
Work Step by Step
We know that the area of a rectangle is given by the following formula: $$A = lw$$ where $l$ is the length and $w$ is the width. We know that the area of this rectangle is $24$ square feet and that the length is two feet greater than its width and can be given by the following equation: $$l = w + 2$$ We plug this information into the formula for the area of a rectangle to find both length and width: $$24 = (w + 2)(w)$$ We can use the distributive property to multiply the monomial with each term in the binomial: $$24 = (w)(w) + (2)(w)$$ Multiply the terms to simplify: $$24 = w^2 + 2w$$ Subtract $24$ from each side of the equation to make the equation equal to zero. Then we can factor to find the values for $w$: $$w^2 + 2w - 24 = 0$$ We want to factor this equation so we can use the zero product property to solve for $w$: To factor, we find the combination of factors for $-24$ that, when added together, equal $2$. Let's examine the factors for $-24$: $-24$ and $1$ or $24$ and $-1$ $-12$ and $2$ or $12$ and $-2$ $-8$ and $3$ or $8$ and $-3$ $-6$ and $4$ or $6$ and $-4$ It looks like $6$ and $-4$ will work because we need a higher positive number than the negative number. Let's factor the equation using the factors we found: $$(w + 6)(w - 4) = 0$$ We use the zero product property, which states that the equation equals zero if either of the factors equals zero. So we set each factor equal to zero: $$w + 6 = 0$$ Subtract $6$ from each side to solve for $w$: $$w = -6$$ Set the other factor equal to zero: $$w - 4 = 0$$ Add $4$ to each side of the equation to solve for $w$: $$w = 4$$ Because measurements cannot be negative, $w$ cannot be $-6$. If $w = 4$, then we can plug this value in for $w$ into the formula for area to solve for $l$: $$24 = (l)(4)$$ Divide each side by $4$ to solve for $l$: $$l = 6$$ The width of the rectangle is $4$ feet and the length is $6$ feet.