the sum of a number and its reciprocal is 61/30. find the number

Question

mathstudent55 · Answer

Start by choosing a variable.
Do you like x?

josefeth00 · Answer

yea I wrote x+1/x = 61/30 but i'm not too sure that's the right way

phi · Answer

that is the correct way.
you can multiply both sides (all terms!) by x
as the next step.

mathstudent55 · Answer

Your equation is correct.
Do you need help solving it?

josefeth00 · Answer

If i multiply all the terms by x, i would get x^2 but what about the 1/x ?

mathstudent55 · Answer

The x's will cancel there.

$\dfrac{1}{x} \times x = \dfrac{1}{x} \times \dfrac{x}{1} = \dfrac{x}{x} = 1$

josefeth00 · Answer

Okay I thought it would be that way I was just very confused. Thanks!

phi · Answer

don't forget to multiply the 61/30 by x also

josefeth00 · Answer

Okay

phi · Answer

what do you have so far?

josefeth00 · Answer

$$x ^{2} -\frac{ 61 }{ 30 }x +1 $$

phi · Answer

=0

josefeth00 · Answer

I will be using quadratic formula to solve for x

phi · Answer

yes.

josefeth00 · Answer

This question is on my textbook and it tells me I cannot use a calculator, is there any easy way to solve without having to multiply so many fractions?

mathstudent55 · Answer

$x \times x + \dfrac{1}{x} \times x  = \dfrac{61}{30} \times x$

$x^2 + 1  = \dfrac{61}{30} x$

$x^2 -  \dfrac{61}{30} x + 1  = 0$

Now continue with:

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

phi · Answer

interesting.  it almost sounds like there is a short-cut.

josefeth00 · Answer

It is easy with a calculator but it is very weird without one

phi · Answer

we could multiply by 30 (both sides, all terms) to get
$$ 30 x^2 -61x +30=0 $$

but that still looks hard to do without a calculator.

phi · Answer

oh, we do get (relatively) nice numbers

josefeth00 · Answer

I got it haha

mathstudent55 · Answer

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