How can I change this y=2/3x^(3/2)-1/2x^(1/2)

  • Context: Undergrad 
  • Thread starter Thread starter gigi9
  • Start date Start date
  • Tags Tags
    Change
Click For Summary
SUMMARY

The discussion focuses on transforming the equation y = (2/3)x^(3/2) - (1/2)x^(1/2) into the form x = ... by eliminating fractions and manipulating the equation. Key steps include multiplying the entire equation by 6, factoring out x^(1/2), and applying the cubic formula to solve the resulting cubic equation 16x^3 - 24x^2 + 9x - 36y^2 = 0. An alternative approach involves substituting z = √x to simplify the equation to 4z^3 - 3z - 6y = 0, which is a reduced cubic form.

PREREQUISITES
  • Understanding of algebraic manipulation and equation transformation
  • Familiarity with cubic equations and the cubic formula
  • Knowledge of factoring techniques, particularly with exponents
  • Basic skills in substitution methods in algebra
NEXT STEPS
  • Study the cubic formula and its applications in solving cubic equations
  • Learn about factoring polynomials, especially those with fractional coefficients
  • Explore substitution methods in algebra, particularly for simplifying complex equations
  • Investigate the properties and applications of cubic functions in mathematics
USEFUL FOR

Students and educators in mathematics, particularly those focusing on algebra and polynomial equations, as well as anyone seeking to improve their skills in equation manipulation and solving cubic equations.

gigi9
Messages
40
Reaction score
0
Help please!

How can I change this y=2/3x^(3/2)-1/2x^(1/2) in term of y, as x=...?
 
Physics news on Phys.org
I hate fractions, so multiply through by 6. Factor out x1/2 on the right. Square each side. Expand the squared terms, distribute the x, subtract 36y2 from each side and apply the cubic formula (I don't know what it is but I know it exists).
 
StephenPrivitera was assuming you meant

y= (2/3)x^(3/2)-(1/2)x^(1/2).

What you wrote could as easily be interpreted as

y= 2/(3x^(3/2)-1/(2x^(1/2))).

Please use parentheses to make your meaning clear.

In a bit more detail, what he said was: multiply the equation by 6 to get
6y= 4x^(3/2)- 3x^(1/2)= (4x- 3)x^(1/2)
so
36y^2= (4x-3)^2(x)= (16x^2- 24x+ 9)x
= 16x^3- 24x^2+ 9x

Which you can write as 16x^3- 24x^2+ 9x- 36y^2= 0 and solve as a cubic equation.

(You can see I have absolutely nothing to do this morning. Well, nothing I want to do!)
 
The easiest approach is not to square each side but instead to subsitute z = \sqrt{x} which leads directly to the reduced cubic of,

4 z^3 - 3 z - 6 y = 0
 
I have to solve the equation 2/3x^(3/2)-1/2x^(1/2) for "X", as x=...
 
Yes, that was what you said before and you got three replies telling you how to do that.
 
Originally posted by HallsofIvy
StephenPrivitera was assuming you meant

y= (2/3)x^(3/2)-(1/2)x^(1/2).

What you wrote could as easily be interpreted as

y= 2/(3x^(3/2)-1/(2x^(1/2))).
I always assume calculator syntax if it's unclear.
 

Similar threads

Undergrad Different Substitution for Solving an Integral
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad ##f(2x)=f^2(x)-2f(x)-1/2## then find ##f(3)##
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Why Is the Integral Result 175/3 Instead of 45?
  • · Replies 4 ·
Replies
4
Views
2K
MHB What is the Slant Asymptote of $\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$?
  • · Replies 3 ·
Replies
3
Views
1K
MHB Implicit function theorem for f(x,y) = x^2+y^2-1
  • · Replies 1 ·
Replies
1
Views
2K
MHB Solve f(x,y,z): Min & Max Values Subject to Constraint x^2+2y^2+6z^2=81
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Integral confusion for a simple Differential Equation
  • · Replies 2 ·
Replies
2
Views
1K
MHB Find the Area of x^2+2x-3 Region
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Problem with function approximation
  • · Replies 12 ·
Replies
12
Views
1K
MHB How to use change of variables technique here?
  • · Replies 2 ·
Replies
2
Views
2K