Problem 5 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$x^{2}=7$$

Step-by-Step Solution

Answer

The roots of the equation are \(x = + \sqrt{7}\) and \(x = - \sqrt{7}\)

Step 1: Isolate the x^2 term

This equation is already in the form \(x^{2}=a\), where a is 7. Therefore there's no further isolation needed at this time.

Step 2: Take the square root of both sides

Take the square root of both sides of the equation, obtaining \(x = ± \sqrt{7}\).

Step 3: Simplify radicals

As 7 cannot be simplified any further, our simplified solutions are \(x = + \sqrt{7}\) and \(x = - \sqrt{7}\).

Step 4: Rationalize denominators

In this equation, there are no denominators to rationalize so this step is not applicable.

Step 5: Final Answer

Therefore, the solutions to the equation \(x^{2}=7\) are \(x = + \sqrt{7}\) and \(x = - \sqrt{7}\).

Key Concepts

Square Root Property

When solving quadratic equations where the variable is squared, such as \(x^2 = a\), the square root property becomes useful. This property tells us that if \(x^2 = a\), then \(x = \pm \sqrt{a}\). The plus-minus sign (\(\pm\)) indicates that there are two possible solutions: one positive and one negative.

• If \(a\) is positive, the solutions are real numbers.
• If \(a\) is negative, the solutions will involve imaginary numbers, since you cannot take the square root of a negative number in the real number system.
• If \(a\) is zero, the solution is simply \(x = 0\).

Using the square root property can simplify solving quadratic equations, particularly when no linear \(x\) term is present. This approach is swift as it avoids the steps involved in methods like factoring or using the quadratic formula.

Simplifying Radicals

To simplify a radical expression like \(\sqrt{7}\), you should aim to find perfect square factors if they exist. However, the number 7 is a prime number, which means it does not have any factors other than 1 and 7 itself.

• If the radical has factors that are perfect squares, simplify by taking the square root of those perfect squares.
• This could turn expressions like \(\sqrt{18}\) into \(3\sqrt{2}\), since \(18 = 9 \times 2\) and \(\sqrt{9} = 3\).
• If there are no perfect square factors, the radical is already in its simplest form, as in our case with \(\sqrt{7}\).

Understanding how to simplify radicals helps not only in clearing up complex expressions but also is essential when working with equations or even graphing functions.

Rationalizing Denominators

Rationalizing denominators is a technique used to eliminate radicals from the denominators of fractions. While this process was not needed in the example equation \(x^2 = 7\), it's a valuable method to know, especially for more complex problems.

• The basic idea is to "clear" the radical by multiplying both the numerator and denominator by an appropriate value.
• For example, if you have \(\frac{1}{\sqrt{2}}\), multiply both top and bottom by \(\sqrt{2}\) to make the denominator rational: \(\frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2}\).
• This process not only aids in finding simple numeric solutions but also is essential for simplifying expressions in calculus and other higher mathematics fields.

Rationalizing denominators can keep solutions consistent and easier to handle, particularly in complex calculations or when graphing.