Problem 33 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ (3 x-4)^{2}=8 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{2\sqrt{2} + 4}{3}\) and \(x = \frac{-2\sqrt{2} + 4}{3}\)

Step 1: Use Square Root Property

The square root property means you can take the square root of both sides of the equation. To achieve this, rewrite the equation as \((3x - 4)^2 = 8\). By implementing this principle, you'll get \(3x - 4 = \sqrt{8}\) or \(3x - 4 = -\sqrt{8}\).

Step 2: Simplify the square root

Simplify the square root. It involves remembering that \(\sqrt{8} = 2\sqrt{2}\). As such, replace the \(\sqrt{8}\) in the equation and get \(3x - 4 = 2\sqrt{2}\) or \(3x - 4 = -2\sqrt{2}\).

Step 3: Solve for x

Isolate the variable \(x\) for both equations by adding 4 to both sides. As such, you'll get: \(3x = 2\sqrt{2} + 4\) or \(3x = -2\sqrt{2} + 4\). Finally, divide by 3 to fully isolate x: \(x = \frac{2\sqrt{2} + 4}{3}\) or \(x = \frac{-2\sqrt{2} + 4}{3}\).

Key Concepts

Square Root Property

Understanding the square root property is crucial when solving quadratic equations like the one given in our exercise, \( (3x - 4)^2 = 8 \). This principle states that if you have an equation of the form \(x^2 = a\), then \(x = \pm\sqrt{a}\). It is used to unravel the squared term when the equation is set up in a way that isolates the squared expresssion on one side.

Applying this property to our example, we take the square root of both sides to eliminate the square on the left side, resulting in two possible solutions: \(3x - 4 = \sqrt{8}\) and \(3x - 4 = -\sqrt{8}\). This step is vital as it acknowledges both the positive and negative square roots of the number on the right-hand side, ensuring that we have considered all possible solutions to the equation.

Isolating Variables

After using the square root property, it's important to isolate the variable to find the values of \(x\) that satisfy the equation. Isolating variables involves performing algebraic operations so that the variable you want to solve for is on one side of the equation, and everything else is on the other.

For our equation, we started with \(3x - 4 = \pm\sqrt{8}\). To isolate \(x\), we first add 4 to each side of both equations resulting in \(3x = \pm2\sqrt{2} + 4\). The next step is to divide each term by 3. By doing this, we get two expressions: \(x = \frac{2\sqrt{2} + 4}{3}\) and \(x = \frac{-2\sqrt{2} + 4}{3}\). At this point, we have isolated \(x\) and now have expressions that can be evaluated or further simplified to determine the actual numerical values of \(x\) that solve the original quadratic equation.

Simplifying Square Roots

Simplifying square roots, as we saw in step two of the solution, involves expressing the square root in its simplest radical form. For instance, \(\sqrt{8}\) is not in simplest form, but \(2\sqrt{2}\) is. This simplification process includes recognizing that \(8 = 2^2 \times 2\) and thus the square root of 8 can be broken down into \(\sqrt{2^2 \times 2}\).

Since the square root of \(2^2\) is 2, we take it out of the square root, leaving us with \(2\sqrt{2}\). This step is important not only to achieve a more refined form of the solution but is also very often the expected way to present the final answer in mathematics. It ensures the result is both accurate and aesthetically pleasing, enhancing clarity and understanding of the solution.