Problem 10 - Precalculus Mathematics for Calculus

Question

Determine whether the given value is a solution of the equation. \(\frac{x^{3 / 2}}{x-6}=x-8\) (a) \(x=4\) (b) \(x=8\)

Step-by-Step Solution

Answer

(a) x = 4 is a solution; (b) x = 8 is not a solution.

Step 1: Substitute x = 4

First, we need to check if \(x = 4\) is a solution to the equation \(\frac{x^{3/2}}{x-6} = x - 8\). Substitute \(x = 4\) into the equation: \[ \frac{4^{3/2}}{4-6} = 4 - 8 \]Calculate \(4^{3/2}\), which is \(\sqrt{4^3} = \sqrt{64} = 8\). Now the equation becomes \(\frac{8}{4-6} = 4-8\). This simplifies to \(\frac{8}{-2} = -4\), which holds true.

Step 2: Analyze x = 8

Now, let's check if \(x = 8\) is a solution. Substitute \(x = 8\) into the original equation: \[ \frac{8^{3/2}}{8-6} = 8 - 8 \]. Compute \(8^{3/2}\), which is \(\sqrt{8^3} = \sqrt{512} = 8 \sqrt{8}\). We simplify \(\frac{8 \sqrt{8}}{2}\), which does not simplify directly to zero because \(8 - 8 = 0\). Thus, this is undefined for this purpose and \(x = 8\) is not a solution.

Key Concepts

Algebra

Algebra is the branch of mathematics that uses symbols, usually letters like \(x\) or \(y\), to represent numbers in equations and expressions. It allows us to solve puzzles or problems by finding unknown values using known relationships. In this exercise, the equation \(\frac{x^{3/2}}{x-6} = x-8\) shows how algebra can be applied to solve for \(x\).
To solve such equations:

• First, understand all parts of the equation. Here, \(x^{3/2}\) represents \(x\) raised to the power of \(1.5\).
• Next, substitute the potential values of \(x\) into the equation.
• Simplify the equation step by step to check if the given value is indeed a solution.

Algebra is powerful because it gives us a systematic way to tackle various mathematical problems.

Radical Expressions

Radical expressions involve roots, such as square roots (\(\sqrt{}\)) or cube roots. They're essential in breaking down expressions that involve fractional exponents. In our example, we encounter \(x^{3/2}\), which can be rewritten using radicals as \(\sqrt{x^3}\).
Let's break it down:

• The exponent \(3/2\) means \(x\) is raised to the third power, and then we take the square root of the result.
• For \(x = 4\), \(4^{3/2}\) becomes \(\sqrt{4^3} = \sqrt{64} = 8\).
• This conversion is crucial to simplify the radical expression into a number you can work with.

Understanding how to manipulate radical expressions is vital for solving complex equations involving fractional exponents.

Solution Verification

Solution verification is a critical step in solving equations, as it ensures that the given or found solution is correct. In this context, we want to see if substituting \(x\) into the equation will satisfy it, thus proving \(x\) is a solution.
Here is the process:

• Substitute the proposed solution, \(x = 4\), into the original equation.
• Simplify each side of the equation to verify if both sides are equal. For instance, substitute \(x = 4\) into \(\frac{x^{3/2}}{x-6} = x-8\); both sides simplify to \(-4\).
• Do the same for any other proposed solutions. If both sides are not equal, like with \(x = 8\), then that value is not a solution.

Through solution verification, we validate our solutions or identify errors, ensuring our results are reliable. This step prevents us from accepting incorrect solutions.