Problem 32 - College Algebra Essentials

Question

Solve each equation in Exercises \(15-34\) by the square root property. $$ (8 x-3)^{2}=5 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x = \frac{3 - \sqrt{5}}{8} \) and \( x = \frac{3 + \sqrt{5}}{8} \)

Step 1: Applying the Square Root Property

In order to utilize the Square Root Property, the equation must be in the form \( (a-b)^2 = c \). Looking at the equation \((8x - 3)^2 = 5\), we can see it's already in that form. Therefore, we apply the Square Root Property, which says that if \( (a-b)^2 = c \) then \( a-b = \sqrt{c} \) or \( a-b = -\sqrt{c} \). Thus, the equation becomes \( 8x - 3 = \sqrt{5} \) or \( 8x - 3 = -\sqrt{5} \).

Step 2: Isolating the variable X

For both equations \( 8x - 3 = \sqrt{5} \) and \( 8x - 3 = -\sqrt{5} \), isolate x by adding 3 to both sides and then dividing by 8. For the first equation, it becomes \( 8x = 3 + \sqrt{5} \) and then \( x = \frac{3 + \sqrt{5}}{8} \). For the second equation, it becomes \( 8x = 3 - \sqrt{5} \) and then \( x = \frac{3 - \sqrt{5}}{8} \).

Step 3: Simplification

No further simplification can be done, so the solutions are \( x = \frac{3 - \sqrt{5}}{8} \) and \( x = \frac{3 + \sqrt{5}}{8} \).

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). One way to solve these equations is by using the square root property. This property is particularly useful for equations that can be expressed in a perfect square form, such as \( (a-b)^2 = c \). By transforming a quadratic equation into this form, you can easily apply the square root property to solve it.
For example, in the given problem \((8x - 3)^2 = 5\), the equation is already in the form \((a-b)^2 = c\). The square root property allows us to take the square root of both sides, resulting in two linear equations:

• \( 8x - 3 = \sqrt{5} \)
• \( 8x - 3 = -\sqrt{5} \)

This yields two potential solutions, requiring further steps to isolate the variable \( x \).

Isolation of Variables

After applying the square root property, the next step is to isolate the variable, often denoted by \( x \). In the context of solving quadratic equations, isolating \( x \) refers to converting an equation into a form where \( x \) stands alone on one side of the equation. This makes finding the solution straightforward.
To isolate \( x \) in the equations \( 8x - 3 = \sqrt{5} \) and \( 8x - 3 = -\sqrt{5} \), follow these steps:

• First, add 3 to both sides of each equation to move the constant term over, resulting in \( 8x = 3 + \sqrt{5} \) and \( 8x = 3 - \sqrt{5} \).
• Then, divide each equation by 8 to solve for \( x \), yielding \( x = \frac{3 + \sqrt{5}}{8} \) and \( x = \frac{3 - \sqrt{5}}{8} \).

Through these steps, we isolate \( x \) to find the solutions to the quadratic equation.

Simplification of Expressions

The simplification step is crucial for presenting the solutions in their simplest and most interpretable form. Simplification refers to the process of reducing mathematical expressions to a form that is easy to analyze or compare. However, sometimes expressions cannot be simplified further while still staying exact and accurate.
In our case, after isolating the variable, we obtain expressions like \( x = \frac{3 + \sqrt{5}}{8} \) and \( x = \frac{3 - \sqrt{5}}{8} \).

• These fractions cannot be simplified further without approximating the square root of 5, which would result in a loss of precision.
• Therefore, they are left in their exact form as the final solution.

Understanding when simplification is complete is key, and recognizing that simplification doesn't always change the appearance of expressions is as important as doing so when necessary.