Problem 26 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ y^{2}=7 $$

Step-by-Step Solution

Answer

Answer: The solutions to the quadratic equation $$y^2 = 7$$ are $$y = \pm\sqrt{7}$$.

Step 1: Rewrite the equation as a root extraction problem

We are given the quadratic equation $$y^2 = 7$$, and we need to find the values of y using the method of extraction of roots. To do this, we will take the square root of both sides of the equation, as shown below: $$\sqrt{y^2} = \sqrt{7}$$

Step 2: Solve for y

Now, we can simply take the square root of both sides of the equation, remembering that we will have both a positive and a negative result: $$y = \pm\sqrt{7}$$

Step 3: Write the final solution

We have found the values of y using the method of extraction of roots. The final solution to the quadratic equation $$y^2 = 7$$ is: $$y = \pm\sqrt{7}$$

Key Concepts

Method of Extraction of Roots

The method of extraction of roots is a straightforward way to solve quadratic equations, especially in forms like \(y^2 = c\) where \(c\) is a constant. The idea is simple: we aim to isolate the variable squared, in this case, \(y^2\), and then "extract" the root by taking the square root of both sides of the equation. This process transforms the equation from a quadratic form to a linear one, making it easier to solve.

Here's how you use the method effectively:

• Identify the part of the equation that has the squared variable. Ensure that it is isolated, meaning the equation should be in the form \(y^2 = c\).
• Apply the square root to both sides. Remember that the square root of \(y^2\) returns \(y\), but importantly, you will typically have two solutions — one positive and one negative.
• The result is \(y = \pm \sqrt{c}\), which gives you both possible roots or solutions.

By following these steps, you can solve simple quadratic equations quickly and effectively using the method of extraction of roots.

Square Root

The square root is a fundamental concept in mathematics. It refers to a number that, when multiplied by itself, gives the original number back. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). When working with square roots in solving equations like \(y^2 = 7\), you apply the square root function to both sides of the equation.

When you take the square root of the square of a variable, such as \(\sqrt{y^2}\), you are essentially finding the original number \(y\) that was squared. In mathematical terms, the square root is often denoted as \(\sqrt{\cdot}\).

Some important points to remember about square roots include:

• Square roots can be both positive and negative because both \(3^2\) and \((-3)^2\) equal 9.
• The symbol \(\pm\) is used to indicate both possible square root values when solving equations.
• Square roots of non-perfect squares, such as 7, are irrational numbers. This means their decimal representation goes on forever without repeating.

Understanding square roots is crucial for manipulating and solving quadratic equations effectively.

Positive and Negative Roots

When solving quadratic equations using the method of extraction of roots, the concept of positive and negative roots becomes very important. It arises because the square of any real number, whether positive or negative, results in a positive number.

For example, both \((+5)^2\) and \((-5)^2\) are equal to 25. Thus, when we take the square root of a positive number, such as \(7\), we must consider both \(\sqrt{7}\) and \(-\sqrt{7}\) as potential solutions. This leads to what are known as positive and negative roots.

Here’s why both values are valid solutions:

• Square roots result from reversing the squaring operation, and since the original number could have been either positive or negative, we account for both possibilities.
• In the context of the equation \(y^2 = 7\), the solution \(y = \pm \sqrt{7}\) tells us that \(y\) could be either \(\sqrt{7}\) or \(-\sqrt{7}\).
• It's crucial to understand that in any quadratic equation, if the variable squared equals a positive number, both the positive and negative solutions are legitimate roots.

Considering both positive and negative roots ensures you find all possible solutions to the quadratic equation.