"Given that f(x)=( 2*x^(3/2) - 3*x^(-3/2) )^2+5, x>0,
a) find, to 3 significant figures, the value of x for which f(x)=5."
I got something entirely different than the book says is the answer (book's answer: 1.14), please help?
and it's c1 so no calcs allowed, i dont get how you do this without a calculator to begin with
+23254 f(x) = ( 2x3/2 - 3x-3/2 )2 + 5
f(x) = 5 ===> ( 2x3/2 - 3x-3/2 )2 + 5 = 5
Subtract 5 from both sides ===> ( 2x3/2 - 3x-3/2 )2 = 0
Find the square root of both sides ===> 2x3/2 - 3x-3/2 = 0
Add 3x-3/2 to both sides ===> 2x3/2 = 3x-3/2
Change the right side ===> 2x3/2 = 3 / x3/2
Multiply both sides by x3/2 ===> 2x3 = 3 (add exponents 3/2 + 3/2 = 6/2 = 3)
Divide both sides by 2 ===> x3 = 3/2
Take the cube root ===> x = 1.447...
Solve for x:
5+(2 x^(3/2)-3/x^(3/2))^2 = 5
Bring 5+(2 x^(3/2)-3/x^(3/2))^2 together using the common denominator x^3:
(4 x^6-7 x^3+9)/x^3 = 5
Multiply both sides by x^3:
4 x^6-7 x^3+9 = 5 x^3
Subtract 5 x^3 from both sides:
4 x^6-12 x^3+9 = 0
Factor 4 x^6-12 x^3+9 into a perfect square:
(2 x^3-3)^2 = 0
Take the square root of both sides:
2 x^3-3 = 0
Add 3 to both sides:
2 x^3 = 3
Divide both sides by 2:
x^3 = 3/2
Taking cube roots gives (3/2)^(1/3) times the third roots of unity:
Answer: | x = -(-3/2)^(1/3) or x = (3/2)^(1/3)=1.145 or x = (-1)^(2/3) (3/2)^(1/3)
