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   x2+1=2.5 x x22.5x+1=0  a=1;b=2.5;c=1 D=b24ac=2.52411=2.25 D>0  x1,2=b±D2a=2.5±2.252 x1,2=1.25±0.75 x1=2 x2=0.5   Factored form of the equation:  (x2)(x0.5)=0 z=1/x1=1/2=12=0.5x + 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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