Problem 37 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ 2 x^{2}=24 $$

Step-by-Step Solution

Answer

Answer: The two possible solutions for the equation \(2x^2 = 24\) are \(x = 2\sqrt{3}\) and \(x = -2\sqrt{3}\).

Step 1: Rewrite the equation

Divide both sides of the equation by 2 to isolate \(x^2\): $$ x^2 = \frac{24}{2} $$ Then simplify the equation: $$ x^2 = 12 $$

Step 2: Extract the square root

Take the square root of both sides of the equation, remembering to include both positive and negative roots: $$ \pm\sqrt{x^2} = \pm\sqrt{12} $$

Step 3: Simplify the expression

Simplify the square root of 12 by factoring out the largest square: $$ \pm\sqrt{4 \cdot 3} = \pm\sqrt{12} $$ Then simplify the square root of 4: $$ \pm2\sqrt{3} = x $$ Now we have the two possible solutions for the equation \(2x^2 = 24\): $$ x = 2\sqrt{3}\text{ or }x = -2\sqrt{3} $$

Key Concepts

Solving Quadratic Equations

Quadratic equations are a central element of algebra and represent polynomials of the second degree, typically in the form \( ax^2 + bx + c = 0 \). The solutions to these equations are the values of \( x \) that make the equation true. There are various methods to solve quadratic equations, such as factoring, completing the square, using the quadratic formula, and the method of extracting roots (also known as the square root method).

The method of extracting roots is particularly useful when the equation is already in a form where \( x^2 \) is isolated. The steps in the provided exercise demonstrate how to apply this method through a sequence of algebraic manipulations to find the values of \( x \) that satisfy the given quadratic equation. Solving quadratic equations is essential in mathematics as it sets the foundation for more complex topics and real-life applications, such as physics, engineering, and economics.

Square Root Method

The square root method offers a straightforward approach when the quadratic equation can be rewritten so that one side has \( x^2 \) and the other side has a constant. The core idea is to apply the principle that if \( x^2 = k \) for some positive number \( k \) , then \( x \) can be either \( \sqrt{k} \) or \( -\sqrt{k} \). This is why, in the given solution, after isolating \( x^2 \) and finding its value to be 12, we take the square root of both sides of the equation, and make sure to consider both the positive and negative roots.

Example of the Square Root Method

The equation \( x^2 = 12 \) can be approached by recognizing that \( 12 \) is not a perfect square, so we proceed by breaking it down into \( 4 \) (which is a perfect square) and \( 3 \) (which is not), resulting in \( x = \pm2\sqrt{3} \). This illustrates the fundamental steps of the square root method and highlights the importance of checking for both positive and negative roots whenever dealing with quadratic equations.

Algebraic Manipulation

Algebraic manipulation is a suite of techniques used to transform mathematical expressions and equations into more manageable or solvable forms. This involves operations like simplifying expressions, factoring, expanding, and isolating variables. In the context of solving quadratic equations, algebraic manipulation allows us to prepare equations for the application of specific solution methods, such as the square root method shown in the example exercise.

Strategies for Algebraic Manipulation

• Combining like terms.
• Factoring common factors.
• Applying the distributive property.
• Using inverse operations to isolate variables.

In the provided exercise, we divided both sides of the equation by 2 to isolate \( x^2 \), which is a form of algebraic manipulation. Simplifying expressions and breaking down numbers into their factors, like we did with \( 12 \) into \( 4 \cdot 3 \) before taking the square root, are also key examples of algebraic manipulation at work. It's a powerful technique that doesn't just help in simple cases but is also fundamental in tackling more complex algebraic problems.