let the two numbers be x and y

x + y = 10

x^{2} + y^{2} = 80

Express x in terms of y

x = 10 - y

(10 - y)^{2} + y^{2} = 80

100 - 20y + y^{2} + y^{2} = 80

100 - 20y + 2y^{2} - 80 = 0

2y^{2} - 20y + 20 = 0

divide through by 2

**y**^{2} - 10y + 10 = 0

we can also formulate another quadratic equation in terms of x by expressing y in terms of x

y = 10 - x

substituting in the second equation

x^{2 }+ (10 - x)^{2} = 80

x^{2} + 100 - 20x + x^{2} = 80

x^{2} + 100 - 20x + x^{2} - 80 = 0

2x^{2} - 20x + 20 = 0

divide through by 2

**x**^{2} - 10x + 10 = 0