Problem 6 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$x^{2}=13$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=\pm\sqrt{13}\)

Step 1: Identify and write down the equation

The given equation is \(x^{2}=13\).

Step 2: Apply the square root property

By applying the square root on both sides of the equation, we get \(x=\pm\sqrt{13}\).

Step 3: Simplify

Since 13 is a prime number and cannot be simplified further, the simplified equation is \(x=\pm\sqrt{13}\).

Key Concepts

Solving Quadratic Equations

When tacking a quadratic equation, like the one in the exercise - \(x^2 = 13\), we're dealing with an equation where the highest power of the variable is 2. The methods for solving these can vary, but one powerful technique is the square root property. This property states that if \(x^2 = a\), then \(x = \pm\sqrt{a}\), where \(\pm\) indicates that there are typically two solutions: one positive and one negative.

To apply this property, you would isolate the squared term (if it’s not already) and take the square root of both sides. In this case, since our equation is already in the form \(x^2 = a\), we can apply the square root to both sides directly, resulting in \(x = \pm\sqrt{13}\). Remember, while this method is quick, it only works when the quadratic equation is in this specific form.

Simplifying Radicals

Upon deriving \(x = \pm\sqrt{13}\), the next step often involves the act of simplifying radicals. A radical is simplified when the radicand (the number under the radical) has no perfect square factors other than 1.

To simplify a radical like \(\sqrt{13}\), you would search for factors of 13 that are perfect squares. However, because 13 is prime, there are no perfect square factors to extract from the radicand. Thus, \(\sqrt{13}\) is already in its simplest form. If, contrary to 13, the radicand had been a number like 18, we could simplify \(\sqrt{18}\) to \(3\sqrt{2}\), as 18 is 9 times 2, and 9 is a perfect square.

Rationalizing Denominators

Occasionally, after solving an equation using the square root property, you might end up with a radical in the denominator of a fraction. To make it more digestible, especially for further calculations, you would use a process called rationalizing the denominator.

This involves multiplying the numerator and the denominator of the fraction by the same radical that is in the denominator. If, for example, you had \(\frac{1}{\sqrt{13}}\), you'd multiply the top and bottom by \(\sqrt{13}\) to eliminate the radical in the denominator, resulting in \(\frac{\sqrt{13}}{13}\). Although it’s not applicable in our initial problem where the solution is \(x = \pm\sqrt{13}\), it's a crucial step when simplifying expressions and ensuring clear and concise answers for algebraic operations.