Problem 70 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ \left(x+\frac{1}{7}\right)^{2}=\frac{11}{49} $$

Step-by-Step Solution

Answer

x = \frac{-1 + \sqrt{11}}{7} or x = \frac{-1 - \sqrt{11}}{7}.

Step 1: Identify the Equation

The given equation is \(\bigg(x + \frac{1}{7}\bigg)^2 = \frac{11}{49}\).

Step 2: Apply the Square Root Property

To remove the square, take the square root of both sides of the equation: \[ \sqrt{\bigg(x + \frac{1}{7}\bigg)^2} = \pm \sqrt{\frac{11}{49}} \].

Step 3: Simplify the Radicals

Simplify \(\sqrt{\frac{11}{49}}\): \[ \sqrt{\frac{11}{49}} = \frac{\sqrt{11}}{7} \]. This simplifies the equation to: \[ x + \frac{1}{7} = \pm \frac{\sqrt{11}}{7} \].

Step 4: Solve for x

To isolate \( x \), subtract \( \frac{1}{7} \) from both sides: \[ x = -\frac{1}{7} \pm \frac{\sqrt{11}}{7} \]. This gives us two potential solutions: \[ x = \frac{-1 + \sqrt{11}}{7} \] and \[ x = \frac{-1 - \sqrt{11}}{7} \].

Key Concepts

square root property

The square root property is a useful tool for solving quadratic equations where the quadratic term can be isolated. When you have an equation of the form \ \(a^2 = b\ \), you can take the square root of both sides to solve for \ a \. This means that \[ a = \pm \sqrt{b} \].
In our example, \left(x+\frac{1}{7}\right)^{2}=\frac{11}{49}\, we first isolate the squared term. Applying the square root property, we get \[ \sqrt{\bigg(x + \frac{1}{7}\bigg)^{2}} = \pm \sqrt{\frac{11}{49}}\].
This property helps simplify certain types of quadratic equations without having to expand and factor them.

simplifying radicals

Simplifying radicals involves breaking down expressions under the square root sign so they are easier to work with. The goal is to express the radical in its simplest form.
In our case, we simplify \[ \sqrt{\frac{11}{49}} \].
To simplify, consider the components inside the square root separately:

• The numerator: \sqrt{11}\ cannot be simplified further because 11 is a prime number.
• The denominator: \sqrt{49} \equals 7, because 7 squared is 49.

Thus, \[ \sqrt{\frac{11}{49}} = \frac{\sqrt{11}}{7} \]. Simplifying radicals allows us to work with a much cleaner expression, which then makes solving for \ x \ much simpler.

algebraic solutions

To find the value of our variable \ x \, we continue solving the equation algebraically. The simplified equation after applying the square root property and simplifying the radicals becomes \ \left( x + \frac{1}{7} \right) = \pm \frac{\sqrt{11}}{7} \.
We need to isolate \ x \. To do that, subtract \frac{1}{7} \ from both sides:
\[ x = -\frac{1}{7} \pm \frac{\sqrt{11}}{7} \],
which results in two potential solutions:

• \[ x = \frac{-1 + \sqrt{11}}{7} \]
• \[ x = \frac{-1 - \sqrt{11}}{7} \]

These algebraic steps show how to isolate the variable and solve the equation completely. Understanding these steps is crucial for mastering solving quadratic equations.