Problem 33 - College Algebra

Question

Solve cach equation in Exercises \(15-34\) by the square root property. $$(3 x-4)^{2}=8$$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Squared Term

To begin with, the squared term \((3x-4)^2\) is already isolated on one side of the equation. The equation is in the required form \(a^2=b\) where, here \(a = 3x-4\) and \(b = 8\).

Step 2: Apply Square Root Property

Next, the square root property states that if \(a^2 = b\), then \(a\) is equal to \(\pm \sqrt{b}\). Thus, we have \(3x-4 = \pm \sqrt{8}\).

Step 3: Simplify the Radical

Now simplify \(\sqrt{8}\) to \(2\sqrt{2}\). So the equation becomes \(3x-4 = \pm 2\sqrt{2}\).

Step 4: Solve for x

Finally, solve for \(x\) by firstly adding 4 to both sides, giving \(3x = 4 \pm 2\sqrt{2}\), then dividing all terms by 3 resulting in \(x = \frac{4 \pm 2\sqrt{2}}{3}\) . This gives us two solutions.

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants. The equation given in the exercise is not in the standard form; it's already set up for using the Square Root Property instead. By recognizing this, we can save time and simplify our approach instantly.

A quadratic can have two, one, or no real solutions depending on the discriminant: the expression under the square root in the quadratic formula \((b^2 - 4ac)\). Here, since our equation simplifies using the Square Root Property, the focus is solely on isolating the squared term and employing the correct algebraic manipulations.

Isolating Squared Terms

When solving quadratic equations using the Square Root Property, it's crucial to first isolate any squared terms. In the exercise, you are given \((3x-4)^2 = 8\). This equation is already isolated, making it straightforward to proceed.

• The term \((3x-4)^2\) is on one side of the equation, while the constant \(8\) is on the other side.
• This conforms to the application condition for the Square Root Property: \(a^2 = b\).

By starting with this isolation, we ensure that we are ready to apply the property and simplify effectively.

Radical Simplification

Radical simplification involves breaking down square roots into their simplest form. In our equation, the task was to simplify \(\sqrt{8}\) which is turned into \(2\sqrt{2}\).

• First, note that \(8 = 4 \times 2\).
• The square root of 4 is 2, hence \(\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\).

Simplifying radicals helps in providing clarity and precision in the solutions, and it is a fundamental skill in algebra that can lead to easier calculations and a clearer final answer.

Algebraic Solutions

After simplifying the radicals, we focus on finding the exact solutions by working algebraically with the equation. The given equation is \(3x - 4 = \pm 2\sqrt{2}\). To solve for \(x\), you perform the following steps:

• Add 4 to both sides: \(3x = 4 \pm 2\sqrt{2}\).
• Divide by 3: \(x = \frac{4 \pm 2\sqrt{2}}{3}\).

This process reveals two potential solutions for \(x\), stemming from the plus-minus operation introduced by the square root property.

Concluding the solution by expressing \(x\) in terms of radicals and constants gives a complete and precise answer. This approach through step-by-step logical operations is fundamental in algebra.