Problem 49 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation. $$27 x^{3}=(x+5)^{3}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{26}\).

Step 1: Recognize the Structure

The given equation is \(27x^3 = (x+5)^3\). Notice that both sides of the equation are perfect cubes, which suggests that we can solve the equation by setting the bases equal since their exponents are the same.

Step 2: Equate the Bases

Since both sides are cubes, we set the bases equal: \(27x = (x+5)\).

Step 3: Simplify the Equation

Now, simplify the equation: \(27x = x + 5\). Subtract \(x\) from both sides to get \(26x = 5\).

Step 4: Solve for \(x\)

Divide both sides by 26 to isolate \(x\): \(x = \frac{5}{26}\).

Step 5: Verify the Solution

To ensure correctness, substitute \(x = \frac{5}{26}\) back into the original equation. Compute both \(27\left(\frac{5}{26}\right)^3\) and \(\left(\frac{5}{26} + 5\right)^3\), and verify they are equal. Simplify each expression: \(27\left(\frac{5}{26}\right)^3 = (x+5)^3 = (\frac{5}{26} + 5)^3\). Both sides should equal \(\frac{15625}{17576}\), confirming the solution is valid.

Key Concepts

Perfect Cubes

Understanding perfect cubes is essential for solving cubic equations like the one in our example: \(27x^3 = (x+5)^3\). A perfect cube is a number that is the result of another number multiplied by itself three times. For instance, \(8\) is a perfect cube because it is \(2^3\), which means \(2 \times 2 \times 2 = 8\). Similarly, \(27\) is a perfect cube since \(3^3 = 27\).
Recognizing the structure of perfect cubes is crucial. In our equation, both sides are expressions raised to the power of three, which means they are perfect cubes. By recognizing this, you can simplify the equation by considering only the bases, because if \(a^3 = b^3\), then \(a = b\). This key property allows us to set \(27x = x+5\) and simplify further.
This simplification enables an easier path to solving the equation, and demonstrates why understanding perfect cubes is a powerful tool in algebra.

Equation Simplification

Simplifying equations involves breaking them down into simpler parts, making them easier to solve. The original equation \(27x^3 = (x+5)^3\) can be daunting at first glance, but simplification is your friend here.
To simplify, we use the property of perfect cubes discussed earlier and set the bases equal: \(27x = x + 5\). This step turns a cubic equation into a linear one, which is much easier to handle.
Here's a simple breakdown of the steps to simplify:

• Start by recognizing the exponents are the same, allowing you to equate the bases.
• Bring all terms involving \(x\) to one side: here you subtract \(x\) from both sides, resulting in \(26x = 5\).
• Once isolated, solve the new linear equation as usual by dividing both sides by 26, giving you \(x = \frac{5}{26}\).

This process highlights how transforming a complex equation into something simpler can help solve it more straightforwardly.

Verification of Solutions

Verifying your solution is a vital step to ensure accuracy. Once you've found \(x = \frac{5}{26}\) as the solution, substitute it back into the original equation to check its validity.
Verification involves a few steps:

• Substitute the solution into the left-hand side of the equation: calculate \(27\left(\frac{5}{26}\right)^3\).
• Do the same for the right-hand side: calculate \(\left(\frac{5}{26} + 5\right)^3\).
• Simplify both expressions and ensure they are equal: both sides should simplify to \(\frac{15625}{17576}\).

This confirms that \(x = \frac{5}{26}\) indeed satisfies the original equation. Verification not only secures the correctness of your solution but also helps reinforce the steps and methods used in solving the problem.