Final answer:
To show that x^2 - 40x - a = 0, we use the perimeter and area formulas for a rectangle. We substitute the expressions for length and width based on the given information, and simplify the equation to obtain x^2 - 40x - a = 0.
Step-by-step explanation:
To show that x^2 - 40x - a = 0, we need to use the given information about the perimeter and area of the rectangle.
The perimeter of a rectangle is given by the formula P = 2(l+w), where l is the length and w is the width.
In this case, the perimeter is 80 cm, so 80 = 2(l+w).
The area of a rectangle is given by the formula A = lw.
Since we don't have the length or width of the rectangle, let's use a and b as placeholders.
So, we have A = ab = a(b) = a(w) = aw.
To relate the perimeter and area, we can use the fact that 80 = 2(l+w), which means l + w = 40.
From this equation, we can solve for l in terms of w, and substitute into the area formula.
l = 40 - w.
A = (40 - w)(w) = 40w - w^2.
Now, we can substitute this expression for A into the equation x^2 - 40x - a = 0.
x^2 - 40x - 40w + w^2 - a = 0.
Simplifying, we get x^2 - (40w + a)x + w^2 = 0.
Since the area equation says A = aw, we can substitute A for aw in the equation.
x^2 - (40w + aw)x + w^2 = 0.
Now we have x^2 - 40wx - awx + w^2 = 0.
Factoring out common factors, we get x(x - 40w) - aw(x - 40w) = 0.
Now we have (x - 40w)(x - aw) = 0.
Since the equation is x^2 - 40x - a = 0, we know that (x - 40w) = x and (x - aw) = -a.
So, we have x - a = 0, which means x = a.
Therefore, x^2 - 40x - a = 0.