Find the exact solution of the equation.

Question

(p ^ (3/2) + p ^ (1/2))/(p ^ (- 1/2)) = 4

Answer 1

Set down:

p^1.5 + p^0.5/p^-0.5 = 4.

p^1.5 + p^{(0.5 − -0.5)} = 4.

p^1.5 + p¹ = 4 or  p^1.5 + p − 4 = 0.

Methods Of Differential Calculus give the First Derivative of p^1.5 + p¹ − 4

 as 1.5p^{(1.5 − 1)} + 1p^{(1 − 1)} − 0 or 1.5p^0.5 + 1p⁰ or 1.5p^0.5 + 1.  

Placing an estimate of p equal to 1.5 into p^1.5 + p gives a result of 3.337117307. This is reasonably close to 4, so adjust p = 1.5 to p = 1.7 and gain 1.7^1.5 + 1.7 or 3.916528818 which lands much closer to 4.

Now place "Function Over Derivative" according to Newton's Method Of Root Approximation and 

build the expression p − [p^1.5 + p − 4]/[1.5p^0.5 + 1]. The table below shows the evaluation of this expression for p = 1.7. Each evaluation is fed back into the expression as "the new p".

p----------------------------------------------p − [p^1.5 + p − 4]/[1.5p^0.5 + 1]

1.7------------------------------------------------------1.728240169

1.728240169-----------------------------------------1.728163202

1.728163202-----------------------------------------1.728163201

1.728163201-----------------------------------------1.728163201

The table finally shows an evaluation of the expression equal to the input p and Newton's Method 

can go no further.

Test p = 1.728163201 in p^1.5 + p to obtain 3.999999999, which verifies an extremely accurate approximation of p for the equation given.