Problem 6 - Precalculus Student Solutions Manual 5th

Question

Match the equation in Column I with its solution \((s)\) in Column II. A. \(\pm 5 i\) B. \(\pm 2 \sqrt{5}\) C. \(\pm i \sqrt{5}\) D. \(5\) E. \(\pm \sqrt{5} \quad\) F. \(-5\) G. \(\pm 5\) H. \(\pm 2 i \sqrt{5}\) $$x^{2}=20$$

Step-by-Step Solution

Answer

B. \( \pm 2 \sqrt{5} \)

Step 1: Identify the equation given

The given equation is \(x^2 = 20\).

Step 2: Solve the equation by isolating x

To solve for x, take the square root of both sides of the equation: \[ x^2 = 20 \implies x = \pm \sqrt{20} \]

Step 3: Simplify the square root

Simplify the expression under the square root: \[ \sqrt{20} = \sqrt{4 \cdot 5} \implies \sqrt{20} = 2 \sqrt{5} \] Thus, \[ x = \pm 2 \sqrt{5} \]

Key Concepts

Equation Simplification

When dealing with quadratic equations like the one in the exercise, simplification is essential. Simplifying an equation helps you isolate the variable and makes solving for it much easier. In this example, we started with the equation \( x^2 = 20 \). Our goal is to simplify it in such a way that we can solve for \( x \) efficiently.

simply put, solving for \( x \) outweighs staying true to the right side of the equation. The more simplified, the cleaner our solution.
To see that at work here, we proceed by taking the square root of both sides: \[ x^2 = 20 \rightarrow x = \sqrt{20} \].
Simplification doesn’t always mean only reducing numbers. Sometimes it means re-writing or altering the equation form to make solving for a variable easier.

Square Roots

Taking the square root is a powerful tool in algebra, especially with quadratic equations. Let's break down the concept here.
For the equation \( x^2 = 20 \), we move forward by taking the square root to isolate \(x\). When you take the square root of both sides of an equation, you end up with:

• The positive square root
• The negative square root

This means: \[ x = \sqrt{20} \text{ or } x = -\sqrt{20} \]. Numbers under the square root can often be simplified for easier understanding. For example, \( \sqrt{20} \) can be rewritten as \( \sqrt{4 \cdot 5} \). Since \( \sqrt{4} = 2 \), this simplifies further to \( 2 \sqrt{5} \).
Always be diligent about checking if you can simplify the square root into more manageable factors.

Solving for x

Finally, let’s piece everything together and solve for \( x \). Our simplified equation now looks like this:
\( x = \pm 2 \sqrt{5} \). This notation \( \pm \) indicates both positive and negative solutions. In this case, we see that \( x \) has two possible values:

• \( 2 \sqrt{5} \)
• \( -2 \sqrt{5} \)

Solving for \( x \) in quadratic equations often results in two solutions because the square of both positive and negative numbers give the same result.
Always recall that there are typically two solutions when you encounter \( x^2 \) results, unless specific constraints limit this.
Reflect on what each part of the original equation tells you about the result and always verify your solution to ensure it works with the original equation.