Find:

Two numbers whose sum -10 and whose product is 5.
;;; x+y = -10 x= -10-y xy=5
(-10-y)y=5 -10y-y^2 = 5
-y^2-10y-5=0
y= -9.47213595499958 and
y= -0.52786404500042
Two numbers whose sum is 1 and whose product is 1.
;;; y= (-1+ sqrt(-3))/-2
y= (-1 - sqrt(-3))/-2

Find the dimensions of a rectangle whose area is 60 cm2 and whose perimeter is 200 cm.

your given solutions are correct. As for the last one,

x+y=100, so
x(100-x)=60
...

To find the dimensions of a rectangle, we need to use the formulas for area and perimeter.

Let's denote the length of the rectangle as L and the width as W.

First, let's consider the given information:

Area of a rectangle = length * width = 60 cm²
Perimeter of a rectangle = 2 * (length + width) = 200 cm

Now, we can set up two equations using these formulas:

Equation 1: L * W = 60 (from the area formula)
Equation 2: 2 * (L + W) = 200 (from the perimeter formula)

We can rearrange Equation 2 to isolate L:

2L + 2W = 200
2L = 200 - 2W
L = 100 - W

Now substitute this value of L into Equation 1:

(100 - W) * W = 60

Expanding and rearranging:

100W - W² = 60
W² - 100W + 60 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. In this case, factoring might be difficult, so let's apply the quadratic formula:

W = (-b ± √(b² - 4ac)) / (2a)

Here, a = 1, b = -100, and c = 60. Substituting these values:

W = (-(-100) ± √((-100)² - 4 * 1 * 60)) / (2 * 1)
W = (100 ± √(10000 - 240)) / 2
W = (100 ± √(9760)) / 2
W = (100 ± 98.798) / 2

Simplifying:

W₁ = (100 + 98.798) / 2 = 99.399
W₂ = (100 - 98.798) / 2 = 1.601

Now, substitute these values back into the equation L = 100 - W to find the corresponding lengths:

L₁ = 100 - 99.399 = 0.601
L₂ = 100 - 1.601 = 98.399

Therefore, the dimensions of the rectangle are:

Length = 0.601 cm, Width = 99.399 cm (or vice versa)
Length = 98.399 cm, Width = 1.601 cm (or vice versa)