Applications: Quadratic equations

Question

Find two positive numbers whose difference is 20 and whose product is 1925

Answer 1

Let x and y be the numbers

Two ways:

x-y=20 --> x = 20+y

xy=1925

(20-y)y = 1925

y²+20y-1925=0

(y-35)(y+55) = 0

y =35,-55 Keep positive number

x=20+y=55

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y=1925/x

x-1925/x=20

x²-20x-1925=0

(x-55)(x+35)=0

x = 55, -35 Keep positive number

y=1925/55=35

Answer 2

x-y = 20 = the difference of two numbers

xy = 1925 = the product of the same two numbers

x = 20+y

(20+y)y = 1925

20y +y^2 = 1925

y^2 +20y -1925 = 0

complete the square

y^2 +20y + 100 = 1925 +100 = 2025

(y+10)^2 = 2025

y+10 =+/- sqr2025 = +/-45

y = -10 +45, or -10-45

y= 35 or -55 but ignore the negative if all you want are positive nubers

x -y =20

x-35 =20

x = 55

the numbers are 35 nd 55