A different approach which does not require a substitution and which does not require having to deal with fractions starts by multiplying both sides by (x − 3)² to get rid of the fraction. We can then proceed as follows x²(x − 3)² + 9x² = 16(x − 3)² x²(x² − 6x + 9) + 9x² = 16(x − 3)² x²(x² − 6x + 18) = 16(x − 3)² (x² − 3(x − 3) + 3(x − 3))(x² − 3(x − 3) − 3(x − 3)) = 16(x − 3)² (x² − 3x + 9)² − 9(x − 3)² = 16(x − 3)² (x² − 3x + 9)² − 25(x − 3)² = 0 (x² − 3x + 9 + 5(x − 3))(x² − 3x + 9 − 5(x − 3)) = 0 (x² + 2x − 6)(x² − 8x + 24) = 0 x = −1 + √7 ⋁ x = −1 − √7 ⋁ x = 4 + 2i√2 ⋁ x = 4 − 2i√2

Nice solution specially how got the (x-3) out of 6x^3+18x^2, neat trick.

Excellent. Straight to the solutions. Well done!

Yet another method is to note that (x − 3)² = x² − 6x + 9 and therefore x² = (x − 3)² + 6x − 9 so we can rewrite the equation x² + (3x/(x − 3))² = 16 as (x − 3)² + 6x − 9 + (3x/(x − 3))² = 16 or (x − 3)² + 6x + (3x/(x − 3))² − 25 = 0 Here (x − 3)² + 6x + (3x/(x − 3))² is a perfect square because 6x is twice the product of (x − 3) and 3x/(x − 3) and we also have 25 = 5² so we get ((x − 3) + 3x/(x − 3))² − 5² = 0 which gives us a difference of two squares at the left hand side whereas the right hand side is zero. This allows us to factor the left hand side using the difference of two squares identity and then apply the zero product property. However, before doing so we first multiply both sides by (x − 3)² to eliminate the fraction so we get ((x − 3)² + 3x)² − (5(x − 3))² = 0 (x² − 3x + 9)² − (5x − 15)² = 0 (x² + 2x − 6)(x² − 8x + 24) = 0 x = −1 + √7 ⋁ x = −1 − √7 ⋁ x = 4 + 2i√2 ⋁ x = 4 − 2i√2

I enjoyed your work and your explanation. Excellent and well done! These problems can be long and it took nearly three pages for this one! Lol Sometimes it feels like you're going down a math rabbit hole with no end but in the end when you do reach a solution (s) it's all good.

Thank you. very nice solved equation.

Never ending marathon mathematics and brain 🧠 puzzled stories ever. Einstein newton and all superb mathematicians brain to be needed to understood not like me the stupid one..not getting any sweet juice extract from your mathematics, all energies i have lost to get it understood sir .really sir you've the genius brain and you're great.❤❤❤after this life i shall pray to God ,make me genius like you.❤❤❤❤❤

Me estoy acordando de mis clases en la prepa…

I prefer to get rid of fraction first, then solution will be of 2 minutes only

It can be solved by converting x to a trigonometric function. (x/4)^2 + (3x/(4(x-3)))^2 = 1 x/4 = cos 3x/(4(x-3)) = sin 16(cs)^2 - 18cs - 9 = 0

Zo‘r

Beautiful

I prefer to take the square root first and then expand with x-3. This is faster.

It's a troubled solution.

Great.

C 1.4 hrs or 84 minutes.

4-

# 30 #

X-x/3-1=0

Почему в ответе Вы сократили на 2..???? Так нельзя!!!

Aare vaai itni si sota math aur kitna lomba korega??

Х=2

I no 😂

Spreco di carta

I want to hear a German 😂

COMPLICADO

Kya bakwass hai yo seedha asnwer ara iska binomial se

1/10..?