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

  • Thread starter Thread starter cloakndagger
  • Start date Start date
cloakndagger
Messages
1
Reaction score
0
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.
 
Mathematics news on Phys.org
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. :)
 

Similar threads

Replies
2
Views
1K
Replies
5
Views
2K
Replies
10
Views
2K
Replies
3
Views
3K
Replies
5
Views
2K
Replies
2
Views
1K
Back
Top