Problem 44 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possib $$(4 x-1)^{2}=15$$

Step-by-Step Solution

Answer

The solutions to the equation are x = 1 and x = -1

Step 1: First step: Move the constant to the other side

Isolate the squared term on one side of the equation, this can be achieved by moving the '-1' term to the other side of the equation. $$ (4 x)^{2} = 15 + 1$$

Step 2: Second step: Extract the square root

The goal is to isolate the term containing the variable x, so the next step is to do the square root to both sides. In this case both \(+\) and \(-\) square roots should be considered as both are possible solutions to the original equation. $$ 4x = \sqrt{16} = \pm 4$$

Step 3: Third step: Solve for x

Finally isolate x by dividing through by 4 on both sides of the equation. $$ x = \pm \frac{4}{4} $$

Step 4: Fourth step: Simplify the solution

Simply the equation to obtain the final answer $$ x = \pm 1 $$

Key Concepts

Isolating squared terms

Isolating squared terms is a crucial step in solving quadratic equations, especially ones that involve a squared variable expression. This process involves rearranging the equation to get the squared expression by itself on one side of the equation. Here's how you can understand it:

• Take the equation \[((4x-1)^2 = 15)\] as given.
• The goal here is to isolate the squared term \((4x-1)^2\) on one side.
• This requires moving any constants or other non-variable terms to the opposite side.

By moving these constants, you set up the equation to make it easier to perform the next mathematical operations. In our example, focus on simplifying it as: \[((4x)^2 = 16)\]. This clear isolation makes it straightforward when transitioning to extracting roots.

Extracting square roots

Extracting square roots is a fundamental operation in solving for variables within quadratic terms, once the squared expression is isolated. In this stage, you take the square root of both sides of the equation. Remember, a square root has two potential answers: a positive and a negative one. For example:

• After isolating the squared term, we have \((4x)^2 = 16\).
• To find 4x, take the square root of each side, giving us \(4x = \pm \sqrt{16}\).
• This simplifies to \(4x = \pm 4\).

It's important to recognize both positive and negative roots here because squaring any number, positive or negative, yields the same positive result. This is critical in ensuring all solutions are accounted for in quadratic equations.

Simplifying irrational solutions

Simplifying solutions is about making solutions to equations as simple as possible—especially when dealing with square roots and irrational numbers. Although in our example, the solution turned out rational, understanding the process for irrational expressions is beneficial. When irrational solutions appear, take these steps:

• If the root expression comes out to an irrational number, you break it down to its simplest form.
• For example, in a situation where the final root calculation yields something like \(\sqrt{50}\), it can be broken down into \(\sqrt{25 \times 2}\), simplifying further to \(5\sqrt{2}\).
• Apply this method of simplifying so that the final answer is clean and easy to use in further calculations.

This approach not only makes the end result neater but also more usable in any subsequent mathematical problems or applications.