Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation by extracting square roots. List both the exact solutions and the decimal solutions rounded to the nearest hundredth. $$9 x^{2}=36$$

Step-by-Step Solution

Answer

The exact solutions are x = 2 and x = -2.

Step 1: Simplify The Equation

We first start by simplifying the equation \( 9 x^{2}= 36 \) by getting the term containing x squared isolated on one side. To do this, divide both sides of the equation by 9 to get \( x^{2} = 4 \)

Step 2: Take The Square Root

The next step is to take the square root of both sides of the equation. Don't forget that we can have both positive and negative square roots. So, we get \( x = ±\sqrt{4} \).

Step 3: Solve for x

Solving for x we find two solutions which are \( x = 2 \) and \( x = -2 \)

Step 4: Rounded Solutions

When already giving the exact solutions as integers, no rounding is needed in this case.

Key Concepts

Extracting Square Roots

Extracting square roots is a valuable technique for solving quadratic equations, especially when the equation is presented in the form of a perfect square, or can be transformed into it. In the exercise given, we start with the equation \(9x^2 = 36\). The first step in solving this equation is to isolate \(x^2\) on one side, which is done by dividing both sides by 9, resulting in \(x^2 = 4\).

At this stage, extracting square roots becomes straightforward. To solve for \(x\), we take the square root of both sides. Remember that square roots have both a positive and negative value. Therefore, the solution becomes \(x = \pm\sqrt{4}\), which ultimately gives us \( x = 2 \) and \( x = -2 \).

This method showcases the simplicity of extracting square roots when dealing with quadratic equations that can be manipulated into a form suitable for this approach. It's especially handy when numbers are perfect squares, making the extraction process intuitive.

Exact Solutions

When we speak about exact solutions, we refer to solutions expressed in their most accurate form, without unnecessary approximation. In this context, the solution to \(x^2 = 4\) after applying the square root gives us exact solutions \(x = 2\) and \(x = -2\).

These solutions are considered exact because they are whole numbers derived from precisely evaluating the square roots of 4. There is no need to express these solutions in another form, as the equation naturally simplifies to integers.

Exact solutions are critical when precision is required or when converting to a decimal form might introduce rounding errors. In these scenarios, ensuring the solutions remain exact maintains the integrity and accuracy of your mathematical findings.

Decimal Solutions

Decimal solutions come into play when we convert exact solutions into a decimal form, often necessary when the results need to be used or presented in a certain context. However, in our exercise, the exact solutions of \(x = 2\) and \(x = -2\) remain the same in decimal form \(2.00\) and \(-2.00\) because they are integers.

In cases where solutions are not perfectly integers, decimals can provide a practical means of expressing results, particularly when precision is essential in real-world applications. For instance, had the square roots not resulted in whole numbers, we would need to round the solutions to the required decimal places.

Having the skills to convert and round decimals appropriately is crucial, especially in scientific calculations where significant digits matter, thus ensuring the effectiveness and accuracy of the solutions adopted.