Second week precal, x^1/2 + 3x^−1/2 = 54x^−3/2

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Discussion Overview

The discussion revolves around solving the equation x1/2 + 3x−1/2 = 54x−3/2. Participants are exploring methods to manipulate and solve this equation, which is part of a precalculus curriculum. The conversation includes attempts to clarify the expressions involved and to guide a participant who is struggling with the setup and solution process.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the problem and seeks help in understanding how to set it up and solve it.
  • Another participant suggests rearranging the equation and factoring out x with the smallest exponent.
  • A clarification is made regarding the interpretation of the term 3x−1/2, correcting an earlier misunderstanding about its meaning.
  • Multiplying the equation by x3/2 is proposed as a method to simplify the equation into a quadratic form.
  • Factoring leads to the expression x−3/2(x + 9)(x - 6) = 0, and the zero-factor property is applied to find potential solutions.
  • There is a discussion about the validity of the solutions obtained, particularly regarding x−3/2 = 0 and whether it yields a real solution.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the solutions, particularly concerning the term x−3/2 = 0. Some participants acknowledge that it does not yield a real solution, while others initially included it in their potential solutions.

Contextual Notes

There are unresolved assumptions regarding the domain of the original equation and the implications of the solutions found, particularly concerning negative values and the nature of the variable x.

cloakndagger
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Im in the first week of Precal and I am running into this problem. I am completely lost as the teacher went way too fast and having trouble even setting up the problem to solve. If someone could walk me thru this I would be grateful. San Jose State Univ Sophmore in Math 19 Precal.

Problem: x^1/2 + 3x^−1/2 = 54x^−3/2

find all real solutions.

I understand that X^1/2 would simply be the square root of X and 3x^-1/2 would be the negative sq root of 3x, but I am soooo completely lost on everything else. no idea how to solve the rest or even set it up.
 
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Hello and welcome to MHB! (Wave)

We are given to solve:

$$x^{\Large{\frac{1}{2}}}+3x^{\Large{-\frac{1}{2}}}=54x^{\Large{-\frac{3}{2}}}$$

I would first arrange as:

$$54x^{\Large{-\frac{3}{2}}}-3x^{\Large{-\frac{1}{2}}}-x^{\Large{\frac{1}{2}}}=0$$

What do you get when you factor out x with the smallest exponent?
 
cloakndagger said:
Im in the first week of Precal and I am running into this problem. I am completely lost as the teacher went way too fast and having trouble even setting up the problem to solve. If someone could walk me thru this I would be grateful. San Jose State Univ Sophmore in Math 19 Precal.

Problem: x^1/2 + 3x^−1/2 = 54x^−3/2

find all real solutions.

I understand that X^1/2 would simply be the square root of X and 3x^-1/2 would be the negative sq root of 3x,
No, '3x^-1/2' is 3 times the reciprocal of the square root of x, 3/x^{1/2}. 'The negative sq root of 3x' would be -(3x)^{1/2}.

but I am soooo completely lost on everything else. no idea how to solve the rest or even set it up.
Multiply the equation by x^{3/2}. That gives x^{1/2+ 3/2)+ 3x^{-1/2+ 3/2}= 54 or x^2+ 3x- 54= 0, a quadratic equation that is easy to solve. (-54= (9)(-6) and 9- 6= 3)
 
i also need help with it
 
Factoring, gives us:

$$x^{\Large{-\frac{3}{2}}}\left(54-3x-x^{2}\right)=0$$

Multiply through by -1 and arrange as:

$$x^{\Large{-\frac{3}{2}}}\left(x^2+3x-54\right)=0$$

Factor quadratic factor:

$$x^{\Large{-\frac{3}{2}}}(x+9)(x-6)=0$$

Using the zero-factor property and observing $x^{\Large{-\frac{3}{2}}}\ne0$, we obtain:

$$x\in\{-9,6\}$$
 
MarkFL said:
...

$$x^{\Large{-\frac{3}{2}}}(x+9)(x-6)=0$$

Using the zero-factor property, we obtain:

$$x\in\{-9,0,6\}$$
$$x^{\Large{-\frac{3}{2}}}=0$$

... has a real solution ?
 
skeeter said:
$$x^{\Large{-\frac{3}{2}}}=0$$

... has a real solution ?

It in fact does not...I should wake up before I begin posting. I will correct my post. :)
 

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