Problem 24 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ b^{2}=6 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation \(b^2 = 6\) using the method of extraction of roots. Answer: The two possible solutions for the given equation are \(b = \sqrt{6}\) and \(b = -\sqrt{6}\).

Step 1: Write the equation

Begin by writing down the given equation: $$ b^2 = 6 $$

Step 2: Extract the roots

Now, we have a quadratic equation with \(b^2\) equal to a constant number. To find the value of \(b\), we can apply the extraction of roots method, which is taking the square root of both sides of the equation. \begin{align*} \sqrt{b^2} &= \pm\sqrt{6} \\ b &= \pm\sqrt{6} \end{align*}

Step 3: Write the final result

From the extraction of roots, we find two possible values for \(b\): $$ b = \sqrt{6},\ -\sqrt{6} $$ So, we have two possible solutions for the given equation: \(b = \sqrt{6}\) or \(b = -\sqrt{6}\).

Key Concepts

Extraction of Roots

Extraction of roots is a straightforward technique often used to solve simple quadratic equations. Quadratic equations, such as \(b^2 = 6\), involve raising a variable to the second power. To solve for the variable, we need to "extract" or "take out" the root of these squares.By extracting the root, we mean applying the square root operation to both sides of the equation to undo the squaring effect. Here, our focus is on balancing the equation by performing the same operation on both sides. When we take the square root of \(b^2\), we don't just get \(b\). Instead, we get \(\pm b\), because both positive and negative numbers, when squared, give the same result. To solve \(b^2 = 6\), we extract the root as follows:

• Calculate \(\sqrt{b^2}\), resulting in \(b\).
• Calculate \(\sqrt{6}\), resulting in \(\pm\sqrt{6}\).

Thus, the values for \(b\) are \(\sqrt{6}\) and \(-\sqrt{6}\). Extracting roots gives us two potential solutions, acknowledging that squaring \(b\) could start with either a positive or negative value.

Square Root

The square root is a fundamental concept in mathematics that relates to the idea of reversing the process of squaring a number. When squaring a number, you multiply it by itself. For instance, if you square \(b\), you get \(b^2\). To find the square root of this result, we need to determine what number multiplied by itself gives the original squared number. This is the essence of the square root operation.Key points about square roots:

• The square root function is denoted by the radical symbol \(\sqrt{.}\).
• Every positive number has two square roots: a positive and a negative.
• The square root of a number \(n\) is anything that, when multiplied by itself, equals \(n\).

In the equation \(b^2 = 6\), finding \(b\) involves using the square root, which means identifying that both \(\sqrt{6}\) and \(-\sqrt{6}\) are solutions. Hence, taking the square root is pivotal in solving such quadratic equations.

Solving Equations

Solving equations is a critical skill in algebra that requires finding the value of the unknown variable that makes the equation true. With quadratic equations like \(b^2 = 6\), solving involves several steps that hinge on understanding algebraic operations.Here's a simplified breakdown:

• **Identify the equation type:** Recognize that \(b^2 = 6\) is a quadratic equation with one variable squared.
• **Utilize appropriate methods:** Decide on the method, such as extraction of roots, to find the variable's solution.
• **Perform calculations:** Apply the method step-by-step, such as taking the square root, to isolate the variable. For \(b^2 = 6\), extract the roots to find \(b = \pm\sqrt{6}\).
• **Verify solutions:** Plug the solutions back into the original equation to ensure that they satisfy it.

Solving equations requires careful application of mathematical methods to systematically arrive at the correct answers. Quadratic equations like these have solutions rooted in such techniques, enabling students to deepen their understanding of algebraic principles.