Problem 21 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$(x+5)^{2}=121$$

Step-by-Step Solution

Answer

The solution to the quadratic equation \((x+5)^{2}=121\) are \(x = 6\) and \(x = -16\).

Step 1: Apply the Square Root Property

The square root property states that if \(k^{2} = p\), then \(k\) is equal to \(\sqrt{p}\) or \(-\sqrt{p}\). Apply this property to the given equation \((x+5)^{2}=121\).

Step 2: Solve for x

First, we take the square root on both sides of the equation, we get \(x+5 = \sqrt{121}\) or \(x+5 = -\sqrt{121}\). The square root of 121 is 11, hence, we get two equations: \(x + 5 = 11\) and \(x + 5 = -11\).

Step 3: Simplify Each Equation

To solve for \(x\) in each equation, we subtract 5 from both sides. Solving these equations gives: \(x = 11 - 5 = 6\) and \(x = -11-5= -16\).

Key Concepts

Square Root Property

The square root property is a fundamental tool in solving quadratic equations when the equation is structured in a squared format. This property states that if you have an equation of the form \[ k^2 = p \] then \( k \) can be either the positive or negative square root of \( p \).
This means:

• \( k = \sqrt{p} \)
• or \( k = -\sqrt{p} \)

Applying the square root property in mathematical problems helps to solve for the unknown variable, especially when it appears in a perfect square form like \[ (x+5)^2 = 121 \] Before applying the square root, ensure the side of the equation containing the square doesn't have any other coefficients or terms. In this example, \( (x+5)^2 \) is isolated, allowing the direct application of the property.
Importantly, always remember that square roots produce both positive and negative results since squaring either value would lead back to the original non-negative number.

Simplifying Radicals

Simplifying radicals is about reducing a square root to its simplest form. This happens when a square root can be expressed without the square root symbol or with smaller numbers under it.
To simplify:

• First, identify if the number under the square root sign is a perfect square.
• If it is, such as \( \sqrt{121} \), simplify it directly to its integer value, 11.
• If it’s not a perfect square, look for factors that are perfect squares and take them out from under the radical.
• Remember, the goal is to express the square root in the simplest terms possible.

This process is critical because it not only makes equations easier to manage but also more understandable for calculations. When you simplify radicals correctly, it can prevent mistakes in further calculations by reducing complex numbers to their simplest versions.

Rationalizing Denominators

Rationalizing the denominator is removing any radicals from the bottom of a fraction. A fraction is considered simplified when its denominator is rational—meaning it doesn’t contain irrational numbers (like square roots).
To rationalize:

• Multiply both the numerator and denominator by a number that will eliminate the radical in the denominator.
• If the denominator is a simple square root, multiply by that square root.
• If it’s more complex, with a binomial denominator such as \(a + \sqrt{b} \), multiply by the conjugate \(a - \sqrt{b} \).
• This process often involves using the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).

Through rationalizing, calculations become more straightforward, and solutions are neater and clearer. This technique is particularly useful in making expressions simpler to evaluate and understand by removing irrational components from the denominator.