# The product

The product of a number plus that number and its inverse is two and one-half. What is the inverse of this number

Result

z =  0.5

#### Solution:

$x + 1/x = 2.5 \ \\ \ \\ \ \\ x^2 + 1 = 2.5 \cdot \ x \ \\ x^2 -2.5x +1 = 0 \ \\ \ \\ a = 1; b = -2.5; c = 1 \ \\ D = b^2 - 4ac = 2.5^2 - 4\cdot 1 \cdot 1 = 2.25 \ \\ D>0 \ \\ \ \\ x_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ 2.5 \pm \sqrt{ 2.25 } }{ 2 } \ \\ x_{1,2} = 1.25 \pm 0.75 \ \\ x_{1} = 2 \ \\ x_{2} = 0.5 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -2) (x -0.5) = 0 \ \\ z = 1/x_{ 1 } = 1/2 = \dfrac{ 1 }{ 2 } = 0.5$

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