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

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The discussion revolves around solving the equation x^(1/2) + 3x^(-1/2) = 54x^(-3/2) in a precalculus context. A user expresses confusion about the problem setup and seeks help after their teacher's fast-paced instruction. The solution involves rearranging the equation and multiplying through by x^(3/2) to simplify it into a quadratic form, x^2 + 3x - 54 = 0. The quadratic can be factored to find the solutions x = -9 and x = 6, but only x = 6 is a valid real solution. The discussion highlights the importance of understanding exponent rules and factoring in solving algebraic equations.
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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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