Problem 81 - Introductory Algebra for College Students

Question

What is the square root property?

Step-by-Step Solution

Answer

The square root property states that if \(x^2 = k\), then \(x\) is equal to either \(\sqrt{k}\) or -\(\sqrt{k}\). It is widely used to solve quadratic equations.

Step 1: Define the square root property

The square root property states that if \(x^2 = k\), where x is a real number and k is a positive number, then \(x = \sqrt{k}\) or \(x = -\sqrt{k}\). This property is derived from the principle that a square of a number is equal to the square of its negative counterpart.

Step 2: Explain the application of the square root property

This property is particularly useful in solving quadratic equations of the form \(x^2 = k\). To find the solution of such equations, it implies to find the square root of \(k\) and its corresponding negative value, as both squaring results back to \(k\). These two roots become the solution.

Step 3: Provide an example of using the square root property

Let's consider an example to illustrate this property effectively. Let's solve the equation \(x^2 = 16\). Using the square root property, we can claim that \(x = \sqrt{16}\) or \(x = -\sqrt{16}\). Hence, \(x\) equals to +4 or -4, since both (+4)^2 and (-4)^2 equal 16.