﻿ Solve the cube root equation \sqrt{x+104} - \sqrt{x-104} = 4

#### Question

Solve the cube root equation

\sqrt{x+104} - \sqrt{x-104} = 4

Collected in the board: Cube root

Steven Zheng posted 4 months ago

Let

a = \sqrt{x+104}
(1)
b = \sqrt{x-104}
(2)

Then

a - b = 4
(3)

Difference of a and b raised to the power of 3 gives

a^3 - b^3 = 208
(4)

Factor LHS using difference of cubes formula

(a - b)(a^2+ab+b^2) = 208
(5)

Substitute (3) to (5) and simplify

a^2+ab+b^2 = 52
(6)

Square of equation (3) gives

(a -b )^2 = 16

Expand LHS

a^2 - 2ab +b^2 = 16
(7)

Substract (7) from (6) and solve for ab

ab = \dfrac{1}{3}(\dfrac{2\cdot 104}{4} - 4^2 ) = 12
(8)

Substituting ab to (6) gives

a^2+b^2 = 52 - ab = 40
(9)

Construct a perfect square binomial with (9) and (8)

(a+b)^2 = a^2+b^2+2ab
(a+b)^2 = 40 +2\cdot 12 = 64

Taking square root of both sides gives

a +b = \pm8
(10)

Now we get two systems of equations

Case 1

\begin{cases} a+b = 8 \\ a - b =4 \end{cases}

which gives

\begin{cases} a= 6 \\ b =6 \end{cases}

Then

x = a^3 - 104 = 6^3 - 104 = 112

Case 2

\begin{cases} a+b = -8 \\ a - b =4 \end{cases}

which gives

\begin{cases} a= -2 \\ b =-6 \end{cases}

Then

x = a^3 - 104 = -2^3 - 104 = -112

In summary, we get two solutions for x.

x = 112 \text{ or }x = -112

Steven Zheng posted 4 months ago

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