Problem 26 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$(x-3)^{2}=15$$

Step-by-Step Solution

Answer

So the solutions to the given equation are \(x = 3 + \sqrt{15}\) and \(x = 3 - \sqrt{15}\)

Step 1: Apply the square root to both sides

The square root property allows for removing the squared aspect of the bracketed part. Apply the square root on both sides of the equation to get \(x - 3 = ± \sqrt{15}\). The ± is because when a square root is removed, the results can be both positive and negative.

Step 2: Isolate x

To solve for \(x\), isolate it by adding 3 to both sides of the equation. This will result in two possible answers, since there's a plus-minus in the equation: \(x = 3 ± \sqrt{15}\).

Key Concepts

Square Root Property

The square root property is a useful technique when solving quadratic equations. When you see a term like \( (x - 3)^2 = 15 \), applying the square root property means taking the square root of both sides of the equation. This effectively removes the square, transforming it into a simpler equation.
There's just one catch: when you take the square root of a number, you end up with two potential solutions, one positive and one negative. That's why we use the "±" sign.
In our example, applying the square root to both sides results in:

• \( x - 3 = ± \sqrt{15} \)

That "±" means \( x - 3 \) could be \( +\sqrt{15} \) or \( -\sqrt{15} \).
The square root property hence helps us break down equations to more manageable forms, providing a pathway to realizing both possible solutions of quadratic equations.

Simplifying Radicals

Simplifying radicals is an important step in dealing with square roots. Not all square roots end up being neat whole numbers, so simplifying them as much as possible is crucial.
For the radical \( \sqrt{15} \), it's important to check if this number can be broken down into smaller, simpler square root terms.
Unfortunately, 15 does not have any perfect square factors other than 1, so \( \sqrt{15} \) is already in its simplest form.
When simplifying radicals, always look to:

• Break down the number under the square root into its prime factors.
• Identify any perfect squares and simplify them out of the radical.

Though \( \sqrt{15} \) cannot be simplified further, being familiar with this process will make it easier to tackle more complex expressions in the future.

Rationalizing Denominators

Rationalizing denominators is a technique often used in simplifying expressions but is not directly needed in solving\( (x - 3)^2 = 15 \).
However, understanding this method is vital when dealing with radicals in a broader context, especially in more complex algebraic manipulations.
Here's what it involves:

• If you have a radical in the denominator (for example, \(1/\sqrt{a}\)), multiply both the numerator and the denominator by the radical to eliminate it from the denominator.
• This process ensures that the denominator is a rational number, hence the term "rationalizing."

Consider \(\frac{1}{\sqrt{2}}\). We can rationalize this by multiplying both top and bottom by \(\sqrt{2}\):\[\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]Thus, the denominator no longer contains a radical. While not essential in all equations, this skill is valuable for further complex algebraic operations and calculus.