Problem 23 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 23-40, solve the quadratic equation by extracting square roots. List both the exact answer and a decimal answer that has been rounded to two decimal places. $$ x^{2}=16 $$

Step-by-Step Solution

Answer

The solution to the equation \(x^{2} = 16\) is \(x = \pm4\). This means x could either be 4 or -4.

Step 1: Isolate x

The first step is to isolate x by taking the square root of both sides of the equation. The equation given is \(x^{2}=16\). So, you start by taking the square root of both sides: sqrt(x^2) = sqrt(16)

Step 2: Simplify

After square roots are taken from both sides of the equation, you get \(x = \pm \sqrt{16}\). This \pm symbol signifies a dual positive or negative value for x.

Step 3: Exact and decimal solutions

\(\sqrt{16}\) equals 4 as exact answer. And also, as derived in step 3, there's a negative value for x as well which is -4. These are the exact solutions. The task, however, also requires a decimal answer, which in this case don't differ from the exact solutions as both are already rounded to two decimal places.

Key Concepts

Extracting Square Roots

The method of extracting square roots is a straightforward strategy for solving specific types of quadratic equations. When you encounter an equation in the form \(x^2 = c\), your aim is to solve for \(x\) by taking the square root of both sides.
This process helps in reducing the quadratic equation to a simpler linear equation. The crucial part is to remember that taking the square root introduces both positive and negative roots. For instance, in the equation \(x^2 = 16\), we calculate \(\sqrt{x^2} = \sqrt{16}\).

• The square root of \(x^2\) simplifies to \(x\).
• The square root of 16 results in both +4 and -4.

Thus, your solution becomes \(x = \pm 4\). This means that \(x\) can be 4 or -4. Understanding and applying this step correctly can greatly aid in simplifying and solving quadratic equations efficiently.

Exact Solutions

An exact solution provides the most precise and simplified form of the answers to a quadratic equation, without any approximations. In the context of solving \(x^2 = 16\), the exact solutions involve evaluating the square root of 16.
Since the square root of 16 is straightforward, the exact solutions are \(x = 4\) and \(x = -4\).

• These solutions are exact because they are not approximated or rounded.
• They provide the true roots of the equation as integers.

The significance of finding exact solutions is that they offer complete accuracy, which can be beneficial in more complex mathematical problems or when further computations are required.

Decimal Solutions

Decimal solutions are derived by rounding the exact answers to a specified number of decimal places. In some cases, particularly when dealing with non-perfect squares, the exact roots are irrational, and decimals can provide a useful approximation.
Nevertheless, in this specific exercise, both exact solutions \(x = 4\) and \(x = -4\) are already decimals at two places: 4.00 and -4.00 respectively.

• It is the nature of the problem that these values do not change when converted to decimals.
• Recognizing when decimal conversion might affect precision is important.

Thus, understanding how to verify your solutions in both exact and decimal forms ensures you can properly interpret and apply these results within context.

Step-by-Step Solutions

A step-by-step solution approach helps break down the problem into manageable parts. This methodical way of solving provides clarity and ensures that nothing important is missed.
Let's look at our example:

• **Step 1**: Start with the given quadratic equation \(x^2 = 16\). The first action is isolating x by taking the square root of both sides.
• **Step 2**: Extract the roots: you simplify to \(x = \pm \sqrt{16}\), which further simplifies to \(x = \pm 4\).
• **Step 3**: Address both the exact solutions (\(4\) and \(-4\)) and their decimal equivalents (concluding they are 4.00 and -4.00).

Applying step-by-step solutions ensures a comprehensive understanding of each phase: isolating, extracting, simplifying, and verifying. This clarity is essential for mastering quadratic equations and for applying these skills to more complex mathematical problems.