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

• MHB
• cloakndagger
In summary, the conversation is about a student struggling with a problem in their Precalculus class and looking for help. The problem involves finding all real solutions to an equation with exponents. The expert suggests factoring and solving a quadratic equation to find the solutions.
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.

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?

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:

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

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:

$$\displaystyle x\in\{-9,6\}$$

MarkFL said:
...

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

Using the zero-factor property, we obtain:

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

... has a real solution ?

skeeter said:
$$\displaystyle x^{\Large{-\frac{3}{2}}}=0$$

... has a real solution ?

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

## What is the meaning of "precal" in the equation x^1/2 + 3x^−1/2 = 54x^−3/2?

Precal is short for precalculus, which is a branch of mathematics that deals with advanced algebra and trigonometry concepts.

## What does the exponent "1/2" represent in the equation x^1/2 + 3x^−1/2 = 54x^−3/2?

The exponent "1/2" represents the square root of the variable x in the equation.

## How do you solve for x in the equation x^1/2 + 3x^−1/2 = 54x^−3/2?

To solve for x, you need to isolate it on one side of the equation. You can do this by using algebraic operations such as multiplication, division, addition, and subtraction. In this equation, you can first multiply both sides by x^3/2 and then simplify to find the value of x.

## What is the significance of the exponent "−1/2" in the equation x^1/2 + 3x^−1/2 = 54x^−3/2?

The exponent "−1/2" represents the reciprocal or the inverse of the square root of x in the equation. It is equivalent to writing 1/x^1/2.

## What are some real-life applications of solving equations like x^1/2 + 3x^−1/2 = 54x^−3/2?

Equations with fractional exponents often arise in physics and engineering problems, such as calculating the velocity of an object in projectile motion or determining the rate of decay of a radioactive substance. They can also be used to solve problems involving compound interest, population growth, and electrical circuits.

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