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Algebraic Fraction Equation

Hello,

I am in the process of completing some maths homework and there was this one question that had me stumped which was:

4/x + x/4 = 5/2

Please can someone talk me through how to solve this equation?
Thanks in advance to anyone who responds!
FLM3026

Reply 1

TheConfusedMedic

cross multiply so you can simplify the LHS to one fraction, and from there you should be able to multiply everything by the denominator and be left with a quadratic which you can solve for x.

Reply 2

Kevin De Bruyne

Original post by FLM3026

What can you do with the LHS?

Reply 3

FLM3026

Original post by surina16

cross multiply so you can simplify the LHS to one fraction, and from there you should be able to multiply everything by the denominator and be left with a quadratic which you can solve for x.

What do I cross multiply, exactly? Do I multiply x/4 by 4/x or do 4/x multiplied by 4?

Reply 4

Zacken

Original post by FLM3026

Whenever you see two fractions being added together, it is usually a good idea to combine them under a single common denominator. In this case, we've got a 4 and an x in the denomiantor, so the common denominator would be 4x. Like so:

\displaystyle \frac{4}{x} + \frac{x}{4} = \frac{4 \times 4}{x \times 4} + \frac{x \times x}{4 \times x} = \frac{16}{4x} + \frac{x^2}{4x} = \frac{16 + x^2}{4x}

.

You've reduced your equation down to

\frac{16+x^2}{4x} = \frac{5}{2}

now. We want to turn this into an equation without any ugly fractions, so we'll multiply both sides by

4x

like this:

\displaystyle \frac{16+x^2}{4x} \times 4x = \frac{5}{2} \times 4x \iff 16+x^2 = 10x \iff \cdots

which is a basic quadratic I'm sure you can solve! Can you see how you can apply this standard method of combining denominators and multiplying (or dividing) both sides of an equation by something to get rid of fractions and turn it into a quadratic or linear or cubic or whatever equation?

Reply 5

TheConfusedMedic

Original post by FLM3026

What do I cross multiply, exactly? Do I multiply x/4 by 4/x or do 4/x multiplied by 4?

a/b + c/d = (ad + bc)/db

Reply 6

FLM3026

Original post by surina16

a/b + c/d = (ad + bc)/db

Oh! Now that rings a bell!

Thank you!
FLM3026 :smile:

Reply 7

FLM3026

Original post by Zacken

\displaystyle \frac{4}{x} + \frac{x}{4} = \frac{4 \times 4}{x \times 4} + \frac{x \times x}{4 \times x} = \frac{16}{4x} + \frac{x^2}{4x} = \frac{16 + x^2}{4x}

.

You've reduced your equation down to

\frac{16+x^2}{4x} = \frac{5}{2}

now. We want to turn this into an equation without any ugly fractions, so we'll multiply both sides by

4x

like this:

\displaystyle \frac{16+x^2}{4x} \times 4x = \frac{5}{2} \times 4x \iff 16+x^2 = 10x \iff \cdots

This makes sense! Thank you for taking the time to draw up this explanation. It's really simple and easy to follow!
Thanks!
FLM3026 :smile:

Reply 8

TheConfusedMedic

Original post by FLM3026

Oh! Now that rings a bell!

Thank you!
FLM3026 :smile:

Great! No problem :h:

Reply 9

Kallisto

Entertainment Forum Helper

Original post by FLM3026

By the look of it, you have to rearrange the equation to a quadratic equation.*

Step one: multiply by 4x on each site of the equation (after doing this you get 16 + x² = 20x/2)
Step two: simplify your equation, if its necessary (you get 16 + x² = 10x)*
Step three: rearrange this equation to a quadratic one by subtracting -10x on each site (you get 16 + x² -10 = 0)
Step four: Sort the equation by grades (you get x² - 10x + 16 = 0)
Step five: simplify the quadratic equation if its necessary.
Step six: calculate the solutions by pq-formula.

That should work. **

Reply 10

Zacken

Original post by FLM3026

This makes sense! Thank you for taking the time to draw up this explanation. It's really simple and easy to follow!
Thanks!
FLM3026 :smile:

No problem.

Reply 11

Kallisto

Entertainment Forum Helper

Original post by Zacken

\displaystyle \frac{4}{x} + \frac{x}{4} = \frac{4 \times 4}{x \times 4} + \frac{x \times x}{4 \times x} = \frac{16}{4x} + \frac{x^2}{4x} = \frac{16 + x^2}{4x}

.

You've reduced your equation down to

\frac{16+x^2}{4x} = \frac{5}{2}

now. We want to turn this into an equation without any ugly fractions, so we'll multiply both sides by

4x

like this:

\displaystyle \frac{16+x^2}{4x} \times 4x = \frac{5}{2} \times 4x \iff 16+x^2 = 10x \iff \cdots

*which is a basic quadratic I'm sure you can solve! Can you see how you can apply this standard method of combining denominators and multiplying (or dividing) both sides of an equation by something to get rid of fractions and turn it into a quadratic or linear or cubic or whatever equation?

May I give you some reputation points for your clarifying way of solution and explanation? nah, just do it.*

Reply 12

Zacken

Original post by Kallisto

May I give you some reputation points for your clarifying way of solution and explanation? nah, just do it.*

Thanks!

Reply 13

Kallisto

Entertainment Forum Helper

Original post by Zacken

Thanks!

You are welcome. You made even an effort to write in latex. That has to be rewarded with reputations. :yep:

Original post by FLM3026

This makes sense! Thank you for taking the time to draw up this explanation. It's really simple and easy to follow!
Thanks!
FLM3026 :smile:

It is so helpful by getting food for thought. Is not that so? wish the best of luck for you in solving another homeworks and works in further subjects.*