Problem 15 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$3 x^{2}-2=0$$

Step-by-Step Solution

Answer

The solutions of the quadratic equation are \(x = \sqrt{6}/3\) or \(x = -\sqrt{6}/3\).

Step 1: Rewrite the equation correctly

Arrange the quadratic equation such that it reads \(ax^{2}=c\). Thus, \(3x^{2}=2\).

Step 2: Apply the square root property

According to the square root property, if \(x^{2}=p\), then \(x=\sqrt{p}\) or \(x=-\sqrt{p}\). Therefore, \(x=\sqrt{2/3}\) or \(x=-\sqrt{2/3}\).

Step 3: Rationalize the denominator

To rationalize the denominator, multiply by the conjugate of the denominator which in this case is \(\sqrt{3}\). Hence, \(x=\sqrt{2/3}\times \sqrt{3}/\sqrt{3} = \sqrt{6}/3\) or \(x=-\sqrt{2/3}\times \sqrt{3}/\sqrt{3} = -\sqrt{6}/3\).

Key Concepts

Understanding the Square Root Property

The square root property is a useful tool when solving quadratic equations. This property states that if you have an equation in the form of \( x^2 = p \), then the solutions are \( x = \sqrt{p} \) and \( x = -\sqrt{p} \). This means you take the square root of both sides of the equation, and remember to include both the positive and negative solutions.

For example, in the equation \( 3x^2 = 2 \), you first need to isolate \( x^2 \) by dividing both sides by 3, resulting in \( x^2 = \frac{2}{3} \). Now, simply apply the square root property to get \( x = \sqrt{\frac{2}{3}} \) and \( x = -\sqrt{\frac{2}{3}} \). The square root property is especially handy because it directly tackles the quadratic term, simplifying the problem significantly.

Simplifying Radicals for Clear Answers

Radicals often appear in solutions to quadratic equations, but they can sometimes be simplified. When you have an expression like \( \sqrt{\frac{2}{3}} \), it might be beneficial to simplify it further.

Simplifying radicals involves looking for perfect square factors in the radicand (the number inside the square root) and simplifying them.

• If the radicand is a fraction, like \( \frac{2}{3} \), it helps to simplify using properties of fractions. However, \( \frac{2}{3} \) doesn't have common factors other than 1, so further simplification involves rationalizing the denominator.
• An example of simplifying a radical could be \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \).

Rationalizing Denominators for Neat Solutions

Rationalizing denominators helps eliminate square roots from the bottom of fractions, making expressions easier to understand and work with.

Take \( \sqrt{\frac{2}{3}} \). Multiplying the numerator and the denominator by \( \sqrt{3} \) will rationalize the denominator:

• Multiply: \( \sqrt{2/3} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{6}}{3} \)
• This process removes the radical in the denominator because \( \sqrt{3} \times \sqrt{3} = 3 \), which is a whole number.

Rationalizing makes expressions clearer and makes further algebraic manipulation easier, especially in complex calculations where clean, whole numbers are preferred.

This is particularly important when presenting final solutions or when using them in further calculations.