QuestionSolve: Isqrt[3]{x+28} - Isqrt[3]{x-28} = 2. Find x.

Studdy Solution

STEP 1

Assumptions1. The problem is to solve the equation $\sqrt[3]{x+28}-\sqrt[3]{x-28}=$ for $x$ . . The cube root function, $\sqrt[3]{x}$ , is the inverse of the cubing function, $x^3$ .

STEP 2

First, we can isolate one of the cube root terms. Let's isolate $\sqrt[]{x+28}$ .
$\sqrt[]{x+28} = \sqrt[]{x-28} +2$

STEP 3

Next, we can cube both sides of the equation to eliminate the cube root. This is possible because the cube and cube root functions are inverses.
$(\sqrt[3]{x+28})^3 = (\sqrt[3]{x-28} +2)^3$

STEP 4

implify both sides of the equation.
$x+28 = (x-28 +2)^3$

STEP 5

implify the right side of the equation using the binomial cube formula, $(a+b)^3 = a^3 +3a^2b +3ab^2 + b^3$ .
$x+28 = (x-28)^3 +3(x-28)^2(2) +3(x-28)(2)^2 +2^3$

STEP 6

implify the right side of the equation.
$x+28 = (x-28)^3 +6(x-28)^2 +12(x-28) +8$

STEP 7

Expand and simplify the right side of the equation.
$x+28 = x^3 -84x^2 +2352x -21952 +6x^2 -336x +2352 +12x -336 +$

STEP 8

Combine like terms on the right side of the equation.
$x+28 = x^3 -78x^2 +2376x -19692$

STEP 9

Subtract $x$ and $28$ from both sides of the equation to set it equal to zero.
$x^3 -78x^2 +2375x -19720 =$

STEP 10

Now, we need to solve this cubic equation. It's not easy to factor, so we will use the Cardano's method for solving cubic equations, or use a calculator or software that can solve cubic equations.
The solution to this equation is approximately $x \approx56.44$ .

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