Answer:
the required equation is x² -10x +18 = 0
Explanation:
Given side x and diagonal 8 cm of a rectangle with perimeter 20 cm, you want to show that the value of x satisfies an equation of the form ...
x² +ax +b = 0 . . . . . where a and b are integers.
Perimeter
The perimeter is the sum of the side lengths. It can be expressed by the formula ...
P = 2(L +W)
Solving for W gives ...
W = P/2 -L
For the given rectangle, the width is ...
W = 20/2 -x = 10 -x
Diagonal
The length of the diagonal is related to the length and width of the rectangle by the Pythagorean theorem.
c² = a² +b²
8² = x² +(10 -x)² = 2x² -20x +100
32 = x² -10x +50 . . . . . . . . divide by 2
x² -10x +18 = 0 . . . . . . . . . . subtract 32
This shows that x must satisfy a quadratic of the form x² +ax +b = 0, where a=-10, and b=18, both integers.
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Additional comment
The vertex form of the equation is ...
(x -5)² -7 = 0
This has solutions ...
x = 5 ±√7 ≈ {2.354, 7.646} . . . . centimeters