the area of a rectangular room is 750 square feet. the width of the room is 5 feet less than the length of…

Explanation:

Step1: Define length and width

Let the length of the room be $y$ feet. Then the width is $y - 5$ feet.

Step2: Use area formula

The area of a rectangle is $A=\text{length}\times\text{width}$. Given $A = 750$ square - feet, so $y(y - 5)=750$.

Step3: Expand the equation

Expand $y(y - 5)$ using the distributive property $a(b - c)=ab-ac$. We get $y^{2}-5y = 750$.

Step4: Rearrange the equation

Rearrange $y(y - 5)=750$ to $750-y(y - 5)=0$.

Step5: Factor the quadratic equation

Starting from $y^{2}-5y - 750=0$, we factor it. We need two numbers that multiply to $-750$ and add up to $-5$. The numbers are $-30$ and $25$. So $y^{2}-5y - 750=(y + 25)(y - 30)=0$.

Answer:

$y^{2}-5y = 750$, $750-y(y - 5)=0$, $(y + 25)(y - 30)=0$