Isolate One Radical. Square Three Times. Terms Cancel. | Math Olympiad

√(x+4)+√(x+5)+√(x+6)=√(x+11) → Find x

Let y=x+5. Equation becomes:
√(y-1)+√y+√(y+1)=√(y+6).

Isolate the pair of radicals:
√(y-1)+√(y+1)=√(y+6)-√y.

Square both sides. The 2y terms cancel:
2√(y²-1)+2√(y²+6y)=6.
√(y²-1)+√(y²+6y)=3.

Square again:
2y²+6y-1+2√((y²-1)(y²+6y))=9.
√((y²-1)(y²+6y))=5-y²-3y.

Square a third time:
(y²-1)(y²+6y)=(5-y²-3y)².

Expand both sides carefully.
The y⁴ and 6y³ terms appear on both
sides and cancel completely.
What remains: -y²-6y=-y²-30y+25.
Simplify: 24y=25. y=25/24.

x=y-5=25/24-5=-95/24.

Verify: √(1/24)+√(25/24)+√(49/24)=√(169/24).
That's (1+5+7)/√24=13/√24. ✓

The three squarings look brutal —
but the cancellation of y⁴ and y³
at the final step turns everything
linear. That collapse is the payoff.

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What's in the video:
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— y=x+5: three consecutive radicals
— Isolating √(y-1)+√(y+1) on left
— First squaring: 2y terms cancel
— √(y²-1)+√(y²+6y)=3
— Second squaring: isolating radical
— Third squaring: y⁴ and y³ cancel
— 24y=25 → y=25/24
— x=-95/24 from back-substitution
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Before you watch:

At the final squaring, expand
(y²-1)(y²+6y) and (5-y²-3y)² fully.

Before simplifying, identify all
the y⁴ terms on both sides and
check that they cancel.

Then do the same for y³.

That double cancellation is what
turns a degree-4 equation linear.

Did you get x=-95/24?
Or did the three squarings put
you off before finishing?
Drop it below.

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Subscribe for radical equations where
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Видео Isolate One Radical. Square Three Times. Terms Cancel. | Math Olympiad канала Math Hacks Hub