Problem 8 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$x^{2}=27$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x=+3\sqrt{3}\) and \(x=-3\sqrt{3}\)

Step 1: Isolate The Square

The first step is to isolate the square or make the equation in the form \((x-a)^2 = b\). In this case, the equation \(x^2=27\) is already in isolated form.

Step 2: Apply The Square Root

Once the square has been isolated, the next step is to apply the square root property. This basically involves taking the square root of both sides. The square root property \(\sqrt{(x-a)^2} = \sqrt{b}\) gives two roots: \(x=a+\sqrt{b}\) and \(x=a-\sqrt{b}\). In this problem, a = 0 and b = 27. Therefore, \(x=+\sqrt{27}\) and \(x=-\sqrt{27}\)

Step 3: Radical Simplification

The radical \(\sqrt{27}\) can be simplified to \(3\sqrt{3}\) because 27 = 9*3 and square root of 9 is 3. So, \(x=+3\sqrt{3}\) and \(x=-3\sqrt{3}\)

Key Concepts

Square Root Property

The square root property is a helpful method for solving quadratic equations, particularly when they take the form \(x^2 = b\). This property states that if \(x^2 = b\), then \(x = \pm \sqrt{b}\). This means the solution to the equation involves both the positive and negative square roots of the number on the other side of the equation. Let's explore its use.

• The first step is to ensure the equation is set up like \(x^2 = b\). Luckily, equations like \(x^2 = 27\) are already set up in this way.
• Next, you apply the square root to each side, which gives you \(x = \pm \sqrt{27}\). This results in two potential solutions: one positive and one negative.

Using the square root property makes solving quadratics straightforward when they're properly isolated. Remember to always consider both the positive and negative roots as solutions.

Radical Simplification

When you solve a quadratic equation by applying the square root property, you might find yourself with a radical expression. Radical simplification involves breaking down the radical to its simplest form.With our example, after applying the square root to 27, we end up with \(\sqrt{27}\).

• First, identify if you can break down the number under the radical into a product of smaller numbers, preferably involving a perfect square.
• In this case, 27 can be broken down as 9 \(\times\) 3. Since 9 is a perfect square, we can simplify this to \(3\sqrt{3}\).

This simplification is crucial because it offers a neater and simpler final solution. Simplified radicals are easier to understand and use in further calculations.

Solving Quadratic Equations Step by Step

Solving quadratic equations step by step using the square root property involves clear and methodical actions.Here's a simple walkthrough of the process:

• Isolate the square: Ensure that the quadratic equation is in the form \(x^2 = b\). If it’s not, rearrange it to fit this pattern.
• Apply the square root property: Once the equation is isolated, take the square root of both sides. This gives you \(x = \pm \sqrt{b}\).
• Simplify the radical: If possible, simplify the radical to its simplest form, following the steps for radical simplification. For instance, simplifying \(\sqrt{27}\) to \(3\sqrt{3}\).

By following these steps methodically, you can effectively solve quadratic equations. This approach makes sure you're handling each part of the equation step by step, minimizing errors and enhancing understanding.