Problem 76 - Algebra 1

Question

Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. $$6 s^{2}-12=0$$

Step-by-Step Solution

Answer

The solutions for the equation are approximately \(1.41\) and \(-1.41\)

Step 1: Factor the Equation

Start out by factoring the equation. Divide equation by 6, it simplifies to \(s^{2} - 2 = 0\)

Step 2: Re-arrange the Equation

The next step is to re-arrange the equation to fit the square root property. So, add 2 to both sides to isolate \(s^2\), which gives \(s^2 = 2\)

Step 3: Apply Square Root Property

Apply the square root property on both sides. Remembering that the square root of a quantity has both negative and positive values, the equation simplifies to \(s = \sqrt{2}\) and \(s = -\sqrt{2}\)

Step 4: Use Calculator

Now, use a calculator to find the decimal value of \(\sqrt{2}\) and round your solutions to the nearest hundredth. \(s \approx 1.41\) and \(s \approx -1.41\)

Key Concepts

Factoring Quadratic Equations

Factoring is a useful method to simplify quadratic equations, making them easier to solve. Quadratic equations are generally in the form of \(ax^2 + bx + c = 0\). In the given problem, we started with the equation \(6s^2 - 12 = 0\). The first step is to factor or divide through the equation by the leading coefficient, which in this case is 6. This simplifies the equation to \(s^2 - 2 = 0\). Factoring not only makes the equation less complex but also prepares it for further solving techniques. Remember that the aim of factoring is to identify a simpler form where variables can be isolated effectively.

Utilizing the Square Root Property

The square root property is a powerful tool for solving equations, especially when we have isolated the squared term on one side of the equation. Once we've simplified the equation to \(s^2 = 2\), we are ready to use this property. The square root property allows us to solve \(s^2 = k\) by taking the square root of both sides. However, it's crucial to remember that every positive number has two square roots: one positive and one negative. Therefore, applying the square root property here gives us two solutions: \(s = \sqrt{2}\) and \(s = -\sqrt{2}\). This dual solution is significant because it highlights the natural symmetry in quadratic equations.

Using a Calculator for Precision

Calculators are invaluable tools when we need exact answers that are cumbersome to compute by hand. After applying the square root property, we found \(s = \sqrt{2}\) and \(s = -\sqrt{2}\). To find these values precisely, use a calculator. Input \(\sqrt{2}\), and the calculator will yield approximately 1.41421356. However, since the problem requires rounding to the nearest hundredth, you would adjust this to 1.41 for \(s = \sqrt{2}\) and similarly \(s = -1.41\) for \(-\sqrt{2}\). Calculators thus not only provide precision but also ensure speed and accuracy in solving complex mathematical equations.

Solving Equations through Systematic Steps

Solving quadratic equations often involves a systematic approach that combines multiple mathematical tools. We started by factoring the given equation to simplify it and then moved to isolating the variable. Next, the square root property helped to derive potential solutions. Finally, we employed a calculator to give precise, rounded answers. This structured method ensures accuracy and clarity, allowing students to handle even seemingly complex equations with confidence.