Problem 58 - Algebra Form and Function

Question

In Problems \(57-59,\) demonstrate a sequence of operations that could be used to solve \(4 x^{2}=16 .\) Begin with the step given. Divide both sides of the equation by \(4 .\)

Step-by-Step Solution

Answer

Answer: The possible values of x are 2 and -2.

Step 1: Divide both sides of the equation by 4

Divide both sides of the equation \(4x^2 = 16\) by 4 to simplify the equation: \[\frac{4x^2}{4} = \frac{16}{4}\]

Step 2: Simplify

After dividing the equation in step 1, we get: \[x^2 = 4\]

Step 3: Find the root

Take the square root of both sides of the equation to isolate x: \[x = \pm\sqrt{4}\]

Step 4: Simplify

Simplify, and we get the final solution: \[x = \pm 2\] Therefore, the possible values of \(x\) are \(2\) and \(-2\).

Key Concepts

Sequence of Operations

Understanding the correct sequence of operations is critical when solving quadratic equations. In our provided exercise, you'll notice that the first step is to divide both sides of the equation by 4. This is a strategic move to isolate and simplify the equation. When faced with an equation like \(4x^2 = 16\), the division step is crucial as it simplifies the problem and makes it more manageable.

Here's how you can think about this sequence of operations:

• First, address the multiplication attached to the variable term (here, it's \(4x^2\)). By dividing both sides by 4, you reduce the equation to a simpler form.
• Upon simplification, you have \(x^2 = 4\). Notice how one step seamlessly leads into another, opening the path to solving for \(x\).
• The final operations include applying a square root to both sides and obtaining the possible values for \(x\), emphasizing the importance of following the steps in order.

Keeping the sequence orderly ensures that each operation supports getting closer to the variable's potential values, minimizing errors and clarifying the solution pathway.

Simplifying Equations

Simplifying equations is often the first action taken when resolving complex algebraic expressions. By reducing the equation \(4x^2 = 16\) through division by 4, we simplify the quadratic equation, making further calculations much easier.

The process of simplification involves these key steps:

• Remove coefficients from terms whenever possible to simplify calculations. Here, dividing by 4 simplifies \(4x^2\) to \(x^2\).
• Clear fractions or larger numbers to reduce computational error.
• Reduce the equation to its simplest form as early as possible in the solution process, allowing a clear view of the next step, finding the root.

This systematic simplification highlights the importance of early-stage reductions in complexity when approaching any quadratic equation or algebraic challenge. It helps to establish a clear path forward in the problem-solving process.

Square Roots

Square roots are essential in solving quadratic equations, especially when you reach a term like \(x^2 = \text{number}\). In the equation \(x^2 = 4\), taking the square root is the crucial step to determine the value or values for \(x\).

Here's why this step is pivotal:

• Applying square roots simplifies an equation from \(x^2\) to \(x\), effectively solving for the variable.
• Remember, when you take the square root of both sides of a square equation, you must consider both the positive and negative roots. Thus, \(\sqrt{4}\) results in \(x = \pm 2\).
• This highlights a characteristic of square roots in quadratic solutions; often, there exist two potential answers due to the nature of squaring negative and positive numbers resulting in the same value (i.e., both \(2^2\) and \((-2)^2\) yield 4).

Keep in mind the dual nature of roots when solving square terms. It's a core concept that ensures you account for all potential solutions in quadratic equations.