Problem 28 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ x^{2}=54 $$

Step-by-Step Solution

Answer

x = \pm 3\sqrt{6}

Step 1: Isolate the quadratic term

The given equation is already in the form where the quadratic term is isolated: \[ x^{2} = 54 \]

Step 2: Apply the square root property

To solve for \( x \), apply the square root property which states that if \( x^{2} = k \), then \( x = \pm \sqrt{k} \). Using this property: \[ x = \pm \sqrt{54} \]

Step 3: Simplify the radical

Simplify \( \sqrt{54} \) by expressing it as a product of simpler square roots. Since \( 54 = 9 \cdot 6 \) and \( 9 \) is a perfect square, you can rewrite the expression: \[ \sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3\sqrt{6} \]

Step 4: Write the final solution

Substitute the simplified radical back into the equation to get the final solution: \[ x = \pm 3\sqrt{6} \]

Key Concepts

Isolating Quadratic Term

Start by isolating the quadratic term in your equation. For the given problem, the equation is:
\[ x^2 = 54 \]
This equation is already in the perfect form where the quadratic term, \( x^2 \), is isolated on one side. This step is crucial because it makes it easier to apply the square root property. Once the quadratic term is alone, you can easily move forward to solve for \( x \).

Simplifying Radicals

After isolating the quadratic term and applying the square root property, you often end up with a radical expression. In this case, after taking the square root of both sides of the equation, you have:
\[ x = \pm \sqrt{54} \]
Simplifying radicals is another key part of solving such equations. Look inside the radical and find factors that are perfect squares. For \( \sqrt{54} \), recognize that 54 can be factored into 9 and 6, where 9 is a perfect square. Therefore,
\[ \sqrt{54} = \sqrt{9 \cdot 6} \ = \sqrt{9} \cdot \sqrt{6} \ = 3\sqrt{6} \]
Thus, the radical \( \sqrt{54} \) simplifies to \( 3\sqrt{6} \). This makes the expression easier to understand and deal with in further calculations.

Solving Quadratic Equations

To solve quadratic equations using the square root property, you should:

• Isolate the quadratic term
• Apply the square root to both sides of the equation
• Simplify the radical if possible

In the given exercise, we start with:
\[ x^2 = 54 \]
Applying the square root to both sides:
\[ x = \pm \sqrt{54} \]
After simplifying the radical:
\[ x = \pm 3\sqrt{6} \]
The final solution provides two possible values for \( x \):
\[ x = 3\sqrt{6} \]
\[ x = -3\sqrt{6} \]
Remember, solving quadratic equations often yields two solutions because you consider both the positive and negative roots.