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

[cloakndagger]

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.

 

[MarkFL]

Hello and welcome to MHB! (Wave)

We are given to solve:

\(\displaystyle x^{\Large{\frac{1}{2}}}+3x^{\Large{-\frac{1}{2}}}=54x^{\Large{-\frac{3}{2}}}\)

I would first arrange as:

\(\displaystyle 54x^{\Large{-\frac{3}{2}}}-3x^{\Large{-\frac{1}{2}}}-x^{\Large{\frac{1}{2}}}=0\)

What do you get when you factor out x with the smallest exponent?

 

[HOI]

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)

 

[konex]

i also need help with it

 

[MarkFL]

Factoring, gives us:

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

Multiply through by -1 and arrange as:

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

Factor quadratic factor:

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

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

\(\displaystyle x\in\{-9,6\}\)

 

[skeeter]

...

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

Using the zero-factor property, we obtain:

\(\displaystyle x\in\{-9,0,6\}\)

\(\displaystyle x^{\Large{-\frac{3}{2}}}=0\)

... has a real solution ?

 

[MarkFL]

\(\displaystyle x^{\Large{-\frac{3}{2}}}=0\)

... has a real solution ?

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