Problem 26 - College Algebra and Calculus: An Applied Approach

Question

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. $$ x^{2}=27 $$

Step-by-Step Solution

Answer

The exact solutions to the quadratic equation \(x^2 = 27\) are \(\sqrt{27}\) and \(-\sqrt{27}\). The decimal approximations, rounded to two decimal places, are 5.20 and -5.20.

Step 1: Identify the Form

The problem is a simple quadratic equation in the form of \(x^2 = a\), where \(a = 27\). This simplifies the task as there is no constant term or coefficient with the x-term in the equation.

Step 2: Extract the Square Root

The next task is to find the square root of 27 to get the value of x. Recall that a quadratic equation has two solutions which are positive and negative square roots. Therefore, \(x = \sqrt{27}\) and \(x = -\sqrt{27}\).

Step 3: Approximate the Square Root

Now find the decimal approximation of \(\sqrt{27}\) and round it to two decimal places. The approximation will give the decimal solutions for the equation.

Key Concepts

Extracting Square Roots

The process of extracting square roots is a classical method used to solve special quadratic equations of the form x^2 = a. In our example, a = 27, indicating that the equation is indeed a perfect candidate for this technique. To solve for x, one needs to determine the principal square root of a, which is the non-negative solution to the equation x^2 = a. However, we must remember that each positive real number actually has two square roots: one positive and one negative. This is significant since quadratic equations inherently possess two solutions.

Therefore, once we've identified the principal square root of a, which is \( \sqrt{27} \), we must also include the negative counterpart, \( -\sqrt{27} \), to account for both potential values of x. It’s crucial for students to realize that omitting the negative solution can lead to incomplete answers. Thus, both \( x = \sqrt{27} \) and \( x = -\sqrt{27} \) are the exact solutions to the given equation.

Quadratic Formula

While our specific exercise did not require the quadratic formula, this method is a powerful tool that provides a generalized solution for all types of quadratic equations, including those that are not simplified to \( x^2 = a \). The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), where \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term of the quadratic equation \( ax^2 + bx + c = 0 \).

The beauty of the quadratic formula lies in its universality. For any quadratic equation, regardless of complexity, this formula can be used to find both real and complex solutions. To utilize this method, one simply substitutes the coefficients of their equation into the formula. This process will yield the solutions once the calculations are completed. It's important for students to be familiar with the quadratic formula, as it is a reliable and systematic approach to finding solutions to any quadratic equation.

Decimal Approximation

When we encounter an equation like \( x^2 = 27 \), we find that the square root of 27 is not a neat integer. In math, especially when dealing with real-world applications, we often need to approximate the exact solutions to a more usable form – typically a decimal. This process is known as decimal approximation.

To approximate \( \sqrt{27} \), we could use a calculator, which might yield a long string of digits. However, for practicality, we round this number to a specified degree of precision, often to one or two decimal places depending on the requirement. In our case, we require a decimal approximation up to two decimal places. Hence, \( \sqrt{27} \) approximates to 5.20 and \( -\sqrt{27} \) approximates to -5.20. It's crucial for students to understand how to round correctly, as different degrees of precision can be asked for in various contexts. Furthermore, students should recognize that while approximations are useful, they must also be aware of the situations wherein exact answers are necessary, such as in pure mathematics or when working with formulas that are sensitive to variance in values.