pls help me :(

Question

4) Find the values of x in the number x, x+1 , x+2 such that the reciprocal of the smallest number equals the sum of the reciprocal of the other two numbers. Assume that x is possitive.

Answer 1

The values of x are x, x+1, x+2

Reciprocal of the smallest number is 1/x

Sum of reciprocal of the other two number is (1/x+1) + (1/x+2)

1/x = (1/x+1) + (1/x+2)

Multiply each side by (x+1)(x+2)

[(x+1)(x+2)] / x = x+2 + x + 1

Multiply each side by x

(x+1)(x+2) = x(2x+3)

x²+x+2x+2 = 2x²+3

x²+3x+2 = 2x²+3

2x²+3- (x²+3x+2) = 0

2x² + 3 - x² - 3x - 2 = 0

x² - 3x+1 = 0

 

Now the expression is the form of Quadratic Equation then values of x are

x = [-(-3) ± √((-3)²-4*1*(1))] / (2*1)

x = [3 ± √(9-4)] / (2)

x = [3 ± √5] / 2

x = [3 ± 2.2360] / 2

x = [3 + 2.2360] / 2 and [3 - 2.2360] / 2

x = 2.6180 and 0.382

Therefore the values of x are 2.6180 and 0.382.