Problem 60 - Algebra 1

Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$3 x^{2}=6$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \sqrt{2}\) and \(x = -\sqrt{2}\)

Step 1: Rearrange the equation

The first step is to isolate the x term. You can do this by dividing both sides of the equation by 3, which gives you \(x^2 = 2\).

Step 2: Solve for x

Next, to determine x, take the square root of both sides of the equation. Remember, when taking the square root of one side, you must include both the positive and negative root, giving \(x = \sqrt{2}\) and \(x = -\sqrt{2}\).

Key Concepts

Isolate the Variable

In the context of solving quadratic equations, one of the essential steps is to 'isolate the variable'. This means you want to get the variable on one side of the equation all by itself. This helps to simplify the equation, making it easier to solve.

To isolate the variable in the equation given, like \(3x^2 = 6\), the first step is to eliminate any coefficients or numbers attached to the variable term. You do this by performing algebraic operations that get you closer to the variable 'x' standing alone. In this case, you divide both sides of the equation by 3, resulting in \(x^2 = 2\), which is a simpler equation with the variable isolated.

Isolating the variable is the pivotal move in simplifying the equation, which paves the way for the subsequent steps of finding the solution. This stage sets the stage for applying further operations, such as extracting square roots, without the clutter of additional numbers or terms.

Square Roots

Once a variable has been isolated in a quadratic equation, we often need to deal with square roots to find the variable's value. A square root, symbolically represented by \(\sqrt{}\), is a value that, when multiplied by itself, gives the original number. For the equation \(x^2 = 2\), we are looking for a number which, when squared, equals 2.

To solve for 'x', we take the square root of both sides of the equation, resulting in \(x = \sqrt{2}\) and \(x = -\sqrt{2}\), since squaring the negative value of \(\sqrt{2}\) also gives 2. It's important to include both the positive and negative roots because squaring either one returns the original value within the equation.

Understanding square roots is key to solving many algebraic equations, not just quadratics. Remember, the square root is the inverse operation of squaring a number and is fundamental in unraveling the values of the variable in the equation.

Radical Expressions

Radical expressions include numbers under the radical sign (like \(\sqrt{2}\)). In our context, after isolating the variable and taking square roots, we express the answers \(x = \sqrt{2}\) and \(x = -\sqrt{2}\) as radical expressions. These expressions represent the exact values of the solutions.

It's possible that you won't always get a neat integer or a rational number after taking the square root. In those cases, leaving the answer in a radical form is the most precise representation. While it might be tempting to use a calculator to express these as decimal numbers, in mathematics, we strive for the most accurate and exact forms of expressions, and radicals serve this purpose well.

Understanding how to manipulate and simplify radical expressions can open doors to solving more complex equations and allow you to maintain the precision of mathematical solutions. Whether through multiplying, dividing, or adding radicals, mastering this concept is a fundamental skill in algebra.