Problem 39 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ 3 x^{2}=5 $$

Step-by-Step Solution

Answer

The solution is \( x = \pm \frac{\sqrt{15}}{3} \).

Step 1: Isolate the variable term

To solve for \( x^2 \), divide both sides of the equation \( 3x^2 = 5 \) by 3.\[ x^2 = \frac{5}{3} \]

Step 2: Solve for x

Take the square root of both sides to solve for \( x \). Remember that taking a square root gives two solutions: one positive and one negative.\[ x = \pm \sqrt{\frac{5}{3}} \]

Step 3: Simplify the square root

Simplify \( \sqrt{\frac{5}{3}} \) by writing it as \( \frac{\sqrt{5}}{\sqrt{3}} \). To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{3} \).\[ x = \pm \frac{\sqrt{15}}{3} \]

Key Concepts

Isolating the Variable

When solving a quadratic equation, one of the first steps you'll need to take is isolating the variable. In our equation, the variable is represented by \( x^2 \). It usually means to get the variable term by itself on one side of the equation, separating it from other numbers and symbols. For the equation \( 3x^2 = 5 \), we want that \( x^2 \) to be alone, so we divide all terms by 3, resulting in \( x^2 = \frac{5}{3} \). This process helps simplify the problem, making it easier to solve in the next steps. Remember, what you do to one side of the equation, you must do to the other to maintain equality.

Rationalizing the Denominator

When working with square roots in fractions, sometimes the denominator has a square root. Mathematically, having a square root in the denominator is not simplified, so we need a method called rationalizing the denominator.To rationalize the denominator in our example \( x = \pm \sqrt{\frac{5}{3}} \), we rewrite it as \( x = \pm \frac{\sqrt{5}}{\sqrt{3}} \). To eliminate the \( \sqrt{3} \) from the denominator, multiply both the numerator and denominator by \( \sqrt{3} \):

• This comes out to: \( \frac{\sqrt{5} \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} \)
• This simplifies to: \( \frac{\sqrt{15}}{3} \)

Now, there are no square roots in the denominator. This makes the expression simplified and easier to interpret.

Simplifying Square Roots

Taking square roots is a common step when solving equations, especially quadratics. However, simplification can often be necessary. In the present example, to simplify \( \sqrt{\frac{5}{3}} \), it's beneficial to break it down:First, understand that \( \sqrt{\frac{5}{3}} \) can be expressed as \( \frac{\sqrt{5}}{\sqrt{3}} \). Then, focus on simplifying each part under the square root.In cases where it applies, look for perfect squares to pull out from under the radical sign. Here, though, neither 5 nor 3 is a perfect square, which is why rationalization helps us.Remember:

• The ultimate goal is making these expressions as simple or elegant as possible.
• Use like terms or recognize if numbers can be further reduced.

Always write your final answer in the most reduced and rationalized form possible.