Problem 62 - Algebra for College Students

Question

Find all real number solutions for each equation. $$3 x^{2}-108=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = -6\).

Step 1: Isolate the Quadratic Term

Start with the equation: \(3x^2 - 108 = 0\). Our first goal is to isolate the term \(x^2\). To do this, add 108 to both sides of the equation: \(3x^2 = 108\).

Step 2: Solve for \(x^2\)

Now we have the equation \(3x^2 = 108\). To solve for \(x^2\), divide both sides by 3: \(x^2 = \frac{108}{3}\). Simplifying the right side gives: \(x^2 = 36\).

Step 3: Take the Square Root

To find \(x\), we take the square root of both sides of the equation \(x^2 = 36\). Remember, the square root of a number can be positive or negative. Thus, \(x = \sqrt{36}\) or \(x = -\sqrt{36}\).

Step 4: Simplify the Square Root

Calculate \(\sqrt{36}\) which is 6. So, \(x = 6\) or \(x = -6\).

Step 5: Conclusion

The real number solutions to the equation are \(x = 6\) and \(x = -6\).

Key Concepts

Real Number Solutions

In the realm of mathematics, when we talk about real number solutions, we are referring to answers to equations that are real numbers. Real numbers are those found on the number line, which include all the integers, fractions, finite decimals, and irrational numbers. Importantly, they do not include complex numbers involving imaginary units. For quadratic equations like the one you're dealing with, the solutions can either be real or complex. In our example problem, since the solutions are both integers, they fall under the category of real solutions. This means they are values that can be plotted on the number line and concretely represent the solutions of our quadratic equation.

Quadratic Term Isolation

Quadratic term isolation is a fundamental step when solving equations like the given one. The initial equation is \(3x^2 - 108 = 0\), and our goal is to isolate the squared term \(x^2\) on one side. This process involves performing algebraic operations to get \(x^2\) on its own.

• The first operation we use is adding 108 to both sides. This step cancels out the -108, resulting in \(3x^2 = 108\).
• Afterwards, to further isolate \(x^2\), we divide the entire equation by 3. This step removes the coefficient of \(x^2\), simplifying the equation to \(x^2 = 36\).

By isolating the quadratic term, we prepare the equation for easier solving in subsequent steps.

Square Root Method

The square root method is an efficient technique for solving equations where the quadratic term has been isolated. Once you have the equation in the form \(x^2 = a\), you can apply the square root method. This involves taking the square root of both sides to solve for \(x\).

• Remember that the square root of any positive number has two values: positive and negative. This is because both a positive and a negative number squared result in the same positive value. For example, \(x^2 = 36\) leads to \(x = \sqrt{36}\) or \(x = -\sqrt{36}\).
• By calculating \(\sqrt{36}\), we find that \(x\) equals 6 or -6. These operations are straightforward but crucial as they provide the solutions to the equations.

The square root method helps quickly move from the isolated quadratic term to finding actual solution values.

Equation Solving Steps

Solving quadratic equations can be systematic when broken down into clear steps. Let’s recap the steps involved using our given equation as an exemplar:

• Step 1: Start with the Original Equation. Our task begins with \(3x^2 - 108 = 0\).
• Step 2: Quadratic Term Isolation. Adjust the equation to get \(x^2\) by itself: \(3x^2 = 108\), then \(x^2 = 36\).
• Step 3: Apply the Square Root Method. Take the square root of both sides: \(x = \sqrt{36}\) and \(x = -\sqrt{36}\).
• Step 4: Simplify and Find the Solutions. Simplifying the square roots, we get \(x = 6\) and \(x = -6\).
• Conclusion: Confirm that the solutions satisfy the original equation, verifying our steps.

By following these steps, you systematically solve the equation, ensuring that no part of the process is skipped or miscalculated.