Problem 40 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \left(x^{3}+56\right)^{1 / 3}=x+2 $$

Step-by-Step Solution

Answer

The solution is \( x = 2 \).

Step 1: Isolate the Cube Root

Subtract 2 from both sides to get the equation in a manageable form: \( \left( x^3 + 56 \right)^{1/3} - 2 = 0 \). This step simplifies the expression for further manipulation.

Step 2: Eliminate the Cube Root

To eliminate the cube root, cube both sides of the equation: \( (x^3 + 56)^{1/3} = x + 2 \). Cubing both sides gives \( x^3 + 56 = (x + 2)^3 \).

Step 3: Expand the Right Side

Expand the right side of the equation: \( (x + 2)^3 = x^3 + 6x^2 + 12x + 8 \).

Step 4: Set the Equation to Zero

Subtract \( x^3 + 56 \) from both sides to obtain \( 0 = 6x^2 + 12x + 8 - 56 \), which simplifies to \( 0 = 6x^2 + 12x - 48 \).

Step 5: Simplify the Quadratic Equation

Simplify the quadratic equation by dividing all terms by 6: \( x^2 + 2x - 8 = 0 \).

Step 6: Solve the Quadratic Equation

Factor the quadratic equation: \( x^2 + 2x - 8 = (x + 4)(x - 2) = 0 \). Set each factor to zero and solve for \( x \): \( x + 4 = 0 \) gives \( x = -4 \), and \( x - 2 = 0 \) gives \( x = 2 \).

Step 7: Verify the Solutions

Plug the solutions back into the original equation for verification. For \( x = 2 \), the original equation holds: \( (2^3 + 56)^{1/3} = 2 + 2 \). For \( x = -4 \), \( ((-4)^3 + 56)^{1/3} = -4 + 2 \) is false, hence invalid.

Key Concepts

Isolating the Cube Root

When solving equations involving a cube root, it's important to first isolate the cube root expression. Doing so allows us to effectively address and manipulate the equation in subsequent steps. In the given equation, \( (x^3 + 56)^{1/3}=x+2 \), the cube root expression is \( (x^3 + 56)^{1/3} \). To isolate it, the goal is to move non-related terms to the opposite side of the equation.

• Subtract 2 from both sides: \( (x^3 + 56)^{1/3} = x + 2 \) becomes \( (x^3 + 56)^{1/3} - 2 = 0 \).
• This isolates the cube root on one side and provides a clear pathway for further handling the equation.

This step sets a foundation for the elimination of the cube root as the next logical strategy.

Eliminating the Cube Root

Once the cube root is isolated, the next step in solving the cubic equation is to eliminate the cube root by cubing both sides. By raising both sides of the equation to the power of three, we rid the equation of the cube root entirely, simplifying it significantly.

• We have \( (x^3 + 56)^{1/3} = x + 2 \). Cubing both sides gives \( x^3 + 56 = (x + 2)^3 \).
• This transformation allows us to work with polynomials instead of radical expressions.

By expanding \( (x + 2)^3 \), you get \( x^3 + 6x^2 + 12x + 8 \).
Thus, we arrive at \( x^3 + 56 = x^3 + 6x^2 + 12x + 8 \). Cancel out \( x^3 \) from both sides to simplify further and prepare to solve the resulting quadratic equation.

Solving Quadratic Equations

After eliminating the cube root and simplifying, a crucial step in the process is solving the quadratic equation you derive. From earlier, we have the equation \( 6x^2 + 12x + 8 - 56 = 0 \), simplifying to \( 6x^2 + 12x - 48 = 0 \).

• Divide every term by 6 to further simplify: this results in \( x^2 + 2x - 8 = 0 \).
• To solve this quadratic equation, you can factor it into \((x + 4)(x - 2) = 0 \).

Setting each factor equal to zero gives us two potential solutions:

• \( x + 4 = 0 \) yields \( x = -4 \).
• \( x - 2 = 0 \) results in \( x = 2 \).

Don't forget to verify these solutions by substituting back into the original equation to ensure they are valid. Here, only \( x = 2 \) satisfies the original equation, confirming it as the solution.