Problem 25 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ k^{2}=14 $$

Step-by-Step Solution

Answer

\( k = \pm \sqrt{14} \)

Step 1 - Isolate the Quadratic Term

The given equation is already in the form needed for the square root property: \[ k^{2} = 14 \]

Step 2 - Apply the Square Root Property

Apply the square root property, which states that if \( k^2 = c \), then \( k = \pm \sqrt{c} \). Therefore, \[ k = \pm \sqrt{14} \]

Step 3 - Simplify the Radicals (if possible)

Check if the radical \( \sqrt{14} \) can be simplified. In this case, 14 does not have any perfect square factors other than 1, so it cannot be simplified further.

Key Concepts

Quadratic Equation

A quadratic equation is any equation that can be written in the form \[ ax^2 + bx + c = 0 \] where a, b, and c are constants, and a is not equal to zero. In this specific problem, we are dealing with a simpler form of a quadratic equation: \[ k^2 = 14 \] Here, the equation only involves the quadratic term without the linear (bx) and constant (c) terms. This simple form makes it easier to solve using certain properties like the square root property. The purpose of solving a quadratic equation is to find the values of the variable that make the equation true.

Simplifying Radicals

Simplifying radicals involves breaking down a radical expression into its simplest form. A radical expression is simplified if there are no perfect square factors other than 1 under the radical sign.
In the context of this exercise, we ended up with \( \sqrt{14} \) after applying the square root property.
To check if \( \sqrt{14} \) can be simplified, we look for any perfect square factors within 14.
Since 14 equals 2 times 7, and neither 2 nor 7 is a perfect square, \( \sqrt{14} \) cannot be simplified any further.

Solving Equations

Solving equations involves finding the value(s) of the variable that make the equation true. When dealing with quadratic equations like \( k^{2} = 14 \), we can utilize the square root property. This property states that if\( k^2 = c \), then\( k = \pm \sqrt{c} \).
Here's a step-by-step approach to this concept:

• First, isolate the quadratic term, which in this exercise is already isolated as \( k^{2} = 14 \).
• Next, apply the square root property by taking the square root of both sides, giving us\( k = \pm \sqrt{14} \).
• Finally, check if the radical can be simplified. In this case, it cannot, so we leave it as\( k = \pm \sqrt{14} \).

These steps help to systematically solve equations, ensuring that all solutions are found and correctly simplified.