Question

@JungHyunRan , the solution to your problem

Answer 1

\(\color{blue}{\text{Originally Posted by}}\) @JungHyunRan
Solve \(\large x^2-5x-2\sqrt{x^2-5x+3}=12\). If the positive real root is in the form of \(\large a+\frac{\sqrt{b}}{c}\), find \(\large (a+4b+c)\)
\(\color{blue}{\text{End of Quote}}\)

Answer 2

First of all substitute:

\[\large{u = x^2-5x}\]

Answer 3

Then your original equation becomes:

\[\large{u-2\sqrt{u+3} = 12}\]

Answer 4

\[\large{\implies u - 12 = 2\sqrt{u+3}}\]

\[\large{\implies (u-12)^2 = 4(u-3)}\]

Answer 5

Simplifying this, we get:

\[\large{u^2 - 28t + 132 = 0}\]

\[\large{\implies (u-6)(u-22) = 0}\]

Answer 6

Now, u = 6 doesn't satisfy the original equation, so only solution is u = 22

Answer 7

Thus,

\[\large{x^2-5x = 22}\]

\[\large{\implies x^2 - 5x - 22 = 0}\]

Answer 8

Now only positive root of this equation is:

\[\large{x = \cfrac{5+\sqrt{113}}{2}}\]

Answer 9

Thus,

\[\large{a = \cfrac{5}{2}}\]

\[\large{b = 113}\]
\[\large{c = 2}\]

Answer 10

Now you can find out a+4b+c

Answer 11

Well there we have it ^^^

\[\large{\color{red}{\text{Solution complete - Vishwesh}}}\]

Answer 12

In case of any doubt feel free to ask and sorry for the delay. I saw your problem about 3 minutes before :)

Answer 13

Are you sure the question asked a+4b+c because usually your answers are integers and this is not ? @JungHyunRan

Answer 14

It may be 4a + something .... Please check that again :)

Answer 15

yeah! I wrote wrong, Sorry :P

Answer 16

Its okay :)

But that won't change any thing as we have a, b and c. :)

Answer 17

Good day !

I have to go now. Leave a message in case of a doubt

Answer 18

yup

Answer 19

Thank you for your help!

Answer 20

Your welcome :)