Simplifying An Interesting Radical

[@wagnerperlino5048]

The second method is much shorter. Congrats !

[@SyberMath]

Thanks! Glad you found it helpful.

[@dmitryli]

My solution was to inspect sqrt(x)+sqrt(x+2) in power of 2.... Once you open the parentheses, you get 2 of what you had in the original expression. That's about it

[@josepherhardt164]

Yeah, somebody pinch me. Actually worked this out ahead of viewing, in only about 5 lines of math. And no, I did not think "Simplifying an Interesting Radical" was about analyzing Jerry Rubin or Timothy Leary. ;)

[@rainerzufall42]

Really enjoyed the second method!

[@SyberMath]

Glad you liked it!

[@rainerzufall42]

@@SyberMath It's quite elegant (and advanced).

[@bobbyheffley4955]

Negative x values yield imaginary numbers.

[@aesthetics_ai]

i felt porud that i had the idea to make it in the beginning +1-1 to make it x^2 +2x +1 - 1 = (x+1)^2 - 1 then consider 1=1^2 thus making ( x+1)^2 - 1^2 = (x+1-1)(x+1+1) and it was all round to x^2 + 2x but yet good for a beginner i guess hehe

[@calculusmethods]

Good methods

[@SyberMath]

Many many thanks

[@scottleung9587]

Nice!

[@Abdurhman_English]

The second method made my head spin 😅.

[@SyberMath]

It's a tough one, but I think it's worth the effort! 😉

[@Christopher-e7o]

X,+2×+5=8

[@Don-Ensley]

problem Simplify √[ x+1+√(x² +2x) ] Let √[ x+1+√(x² +2x) ] = √a + √b Square. x+1+√(x² +2x) = a+b+√(4ab) Make a system of equations. x + 1 = a + b x² + 2 x = 4 a b b = x + 1 - a Substitute into second equation. x² + 2 x = 4 a (x + 1 - a) Solve for a. x² + 2 x = 4 a x + 4 a - 4 a² Rearrange into standard form as a quadratic in a. 4a²-4(x+1) a + x² + 2x = 0 Use the quadratic formula. a = { 4(x+1) ± 4 } / 8 = { (x+1) ± 1 } / 2 = { (x+1) ± 1 } / 2 = (x + 2)/2, x / 2 b = x + 1 - a = x + 1 -(x + 2)/2, x + 1 - x/2 = x / 2, ( x + 2 ) / 2 √[ x+1+√(x² +2x) ] = √a + √b Since addition is symmetric, the radical simplifies to √[ x+1+√(x² +2x) ] = √(x / 2) + √[(x + 2)/2] = [√ x + √ (x+2)]/ √2 answer √[ x+1+√(x² +2x) ] = √(x/2) + √(x/2+1)

[@StaR-uw3dc]

√(x+1+√(x²+2x)) = √(2x+2+2√(x(x+2)))/√2 = √(x+x+2+2√x√(x+2))/√2 = √(√x+√(x+2))²/√2 = (√x+√(x+2))/√2 = (√(2x)+√(2x+4))/2

[@bayareapianist]

You wasted so much writing. All you had to do was to square both sides.

[@Abdurhman_English]

@@bayareapianist This is an expression not an equation.

[@bayareapianist]

@Abdurhman_English watch the solution #2