Problem 46 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ 2 x^{2}=9 $$

Step-by-Step Solution

Answer

\( x = \pm \frac{3 \sqrt{2}}{2} \)

Step 1: Isolate the quadratic term

Divide both sides of the equation by 2 to isolate the quadratic term: \[ \frac{2 x^2}{2} = \frac{9}{2} \] Simplifying: \[ x^2 = \frac{9}{2} \]

Step 2: Apply the square root property

Take the square root of both sides to solve for \( x \): \[ x = \pm \sqrt{\frac{9}{2}} \]

Step 3: Simplify the radical

Write the square root as a product of two square roots and simplify: \[ x = \pm \sqrt{\frac{9}{2}} = \pm \frac{\sqrt{9}}{\sqrt{2}} = \pm \frac{3}{\sqrt{2}} \] Rationalize the denominator: \[ x = \pm \frac{3 \sqrt{2}}{2} \]

Key Concepts

Square Root Property

The square root property is a useful tool for solving quadratic equations of the form \[ x^2 = k \]. It states that if you have \[ x^2 = k \], then \[ x = \pm \sqrt{k} \]. This approach allows us to not only find the positive root but also the negative root, which is essential because quadratic equations often have two answers. Here's how it applies to our exercise:

After isolating the quadratic term, we get \[ x^2 = \frac{9}{2} \]. By applying the square root property, we take the square root of both sides to solve for \[ x \]:

\[ x = \pm \sqrt{\frac{9}{2}} \].

This step gives us two potential solutions: the positive and negative square roots. Remember, the square root property is a quick way to both isolate \[ x \] and get all possible solutions to the equation.

Simplifying Radicals

After applying the square root property, the next step involves simplifying the resulting radical expression. Simplifying radicals makes equations easier to understand and work with. For the expression \[ \sqrt{\frac{9}{2}} \], we break it down into simpler parts:

First, use the property of square roots that states \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \].

For our example: \[ x = \pm \sqrt{\frac{9}{2}} = \pm \frac{\sqrt{9}}{\sqrt{2}} \].

Then, simplify the square roots separately. Since \[ \sqrt{9} = 3 \], we get: \[ x = \pm \frac{3}{\sqrt{2}} \].

Remember, always simplify roots whenever possible. It makes further operations easier and the solution more elegant.

Rationalizing the Denominator

Rationalizing the denominator is crucial for having cleaner expressions. It means converting a fraction into an equivalent form so that the denominator does not contain any radicals. When we reached \[ x = \pm \frac{3}{\sqrt{2}} \], we needed to rationalize the denominator:

Multiply the numerator and the denominator by \[ \sqrt{2} \] to get:

\[ x = \pm \frac{3 \sqrt{2}}{\sqrt{2} \sqrt{2}} \]. This simplifies to: \[ x = \pm \frac{3 \sqrt{2}}{2} \].

Now, the denominator is a simple integer, which makes the expression nicer and more usable. This step helps ensure that our solutions are presented in the standard mathematical form. Practice rationalizing denominators with other radicals to become more comfortable with the process!