Question

Solving the equation with three radical terms

\sqrt{x-1}+\sqrt{2x-3} +\sqrt{3x-5} = 3x-3

Collected in the board: Radical equations

Steven Zheng posted 3 days ago

Answer

\sqrt{x-1}+\sqrt{2x-3} +\sqrt{3x-5} = 3x-3

Using the substitution method for the equation with more than three radicals

Let

a = \sqrt{x-1}
(1)
b = \sqrt{2x-3}
(2)
c = \sqrt{3x-5}
(3)

Then

a+b+c = 3x-3
(4)
a^2+b^2+c^2 = x-1+2x-3+3x-5 = 6x-9
(5)

Subtracting 2 times of (4) from (5) gives

a^2+b^2+c^2 - 2a-2b-2c = -9+6 = -3

Then

(a-1)^2+(b-1)^2+(c-1)^2 = 0

To make the equation hold true, if and only if

a = 1

b=1

c=1

Substitute the values of a,b,c to (1), (2), (3) and solve for x

1 = \sqrt{x-1} \implies x = 2

1 = \sqrt{2x-3} \implies x = 2

1 = \sqrt{3x-5} \implies x = 2

Therefore

The equation has only 1 solution x = 2

Steven Zheng posted 3 days ago

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