Problem 97 - A Graphical Approach to Precalculus with Limits

Question

Incorporate many concepts from earlier work with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ What are the three proposed solutions of the original equation, $$\sqrt[3]{4 x-4}=\sqrt{x+1} ?$$

Step-by-Step Solution

Answer

The proposed solutions are \( x = 1 \), \( x = 3 \), and \( x = 5 \).

Step 1: Eliminate the roots

Raise both sides of the equation to the power of 3 to eliminate the cube root on the left side:\[ (\sqrt[3]{4x - 4})^3 = (\sqrt{x + 1})^3 \]This simplifies to:\[ 4x - 4 = (x + 1)^{3/2} \].

Step 2: Simplify the equation

Next, square both sides to eliminate the square root on the right side:\[ (4x - 4)^2 = (x + 1)^3 \]Expanding both sides gives:\[ (4x - 4)(4x - 4) = (x + 1)(x + 1)(x + 1) \].

Step 3: Further simplify and solve

Expand the left side:\[ 16x^2 - 32x + 16 \]And the right side:\[ x^3 + 3x^2 + 3x + 1 \].

Step 4: Set the equation to zero

Bring all terms to one side to form a polynomial equation:\[ 16x^2 - 32x + 16 - x^3 - 3x^2 - 3x - 1 = 0 \].

Step 5: Arrange and solve the polynomial equation

Combine like terms:\[ -x^3 + 13x^2 - 35x + 15 = 0 \].

Step 6: Find roots of the polynomial

Using techniques such as synthetic division or a calculator, solve the polynomial equation. Proposed solutions are found to be \( x = 1 \), \( x = 3 \), and \( x = 5 \).

Step 7: Verify each solution

Check each solution in the original equation to confirm they satisfy the equation. Upon verification, it shows that all three values satisfy the original equation.

Key Concepts

Polynomial Equations

Polynomial equations are foundational in algebra and involve expressions with variables raised to whole number powers.
The general form of a polynomial equation is:

• \[ a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0 \]

where \( n \) is a non-negative integer, and \( a_n, \ldots, a_0 \) are constants with \( a_n eq 0 \).
In the context of solving radical equations like the one given, converting the equation into a polynomial is a crucial step.
After eliminating the roots by raising both sides to appropriate powers, we often end up with a polynomial equation. These are easier to handle and solve.
Steps such as combining like terms are employed to simplify the equation further, preparing it for solution methods like synthetic division.
Some key points about polynomial equations include:

• The highest exponent of the variable (called the degree) determines the number of roots and the overall behavior of the polynomial.
• Methods like factoring, synthetic division, and using the Rational Root Theorem can help find the solutions to polynomial equations.

The problem here illustrates turning a complex radical equation into a simple polynomial equation: \[ -x^3 + 13x^2 - 35x + 15 = 0 \] This specific third-degree equation is then tackled to find its roots.

Cube Roots

The cube root of a number \( a \) is a value \( x \) such that \( x^3 = a \). In equations like \( \sqrt[3]{4x - 4} = \sqrt{x+1} \), cube roots are involved on one side.
Applying appropriate operations can eliminate these roots, simplifying the problem.
To handle an equation with cube roots generally involves several steps:

• **Raise both sides to the third power:** This eliminates the cube root, restoring the expression within to its original form, e.g.: \[ (\sqrt[3]{b})^3 = b \]

This step must be done carefully to avoid mistakes, as any calculation mistake affects all later work.
Practical skills in managing cube roots can be beneficial not only in academic exercises but also in fields requiring mathematical literacy.
Remember: when dealing with cube roots and their elimination:

• Be mindful of both sides of the equation. Operations should always balance to maintain equality.
• Watch for potential extraneous solutions introduced by these operations.

Verification of Solutions

After arriving at possible solutions from a polynomial or other equation, it's essential to verify each one to ensure they satisfy the original equation.
This step confirms the correctness and relevance of the solutions.
Here's how verification works:

• Plug each potential solution back into the original equation.
• Calculate both sides to confirm if the equality holds true.

Verification in our case involves substituting each of the proposed solutions \( x = 1 \), \( x = 3 \), and \( x = 5 \) back into the original radical equation:
\( \sqrt[3]{4x - 4} = \sqrt{x+1} \)

• For \( x = 1 \): Check if both sides equal by calculating \( \sqrt[3]{4 \cdot 1 - 4} \) and \( \sqrt{1 + 1} \).
• Repeat for each \( x = 3 \) and \( x = 5 \).

In practical terms, verification helps to catch:

• **Errors made during calculations**: It's a double-check for correctness, a vital step in any problem-solving process.
• **Extraneous solutions**: Solutions that might satisfy transformed versions but not the original equation.

This ensures only the true solutions are considered valid, emphasizing accuracy in mathematical practice.