# Difference of two number

The difference of two numbers is 20. They are positive integers greater than zero. The first number raised to one-half equals the second number. Determine the two numbers.

Result

a =  25
b =  5

#### Solution:

$a-b = 20 \ \\ b^2 = a = 25 = 0 \ \\ \ \\ b^2-b-20 = 0 \ \\ b^2 -b -20 = 0 \ \\ \ \\ p = 1; q = -1; r = -20 \ \\ D = q^2 - 4pr = 1^2 - 4\cdot 1 \cdot (-20) = 81 \ \\ D>0 \ \\ \ \\ b_{1,2} = \dfrac{ -q \pm \sqrt{ D } }{ 2p } = \dfrac{ 1 \pm \sqrt{ 81 } }{ 2 } \ \\ b_{1,2} = \dfrac{ 1 \pm 9 }{ 2 } \ \\ b_{1,2} = 0.5 \pm 4.5 \ \\ b_{1} = 5 \ \\ b_{2} = -4 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (b -5) (b +4) = 0b = b_{ 1 } = 5 \ \\ a = b^2 = 5^2 = 25$

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$b = b_{ 1 } = 5$

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