Problem 27 - 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. $$(x-12)^{2}=16$$

Step-by-Step Solution

Answer

The exact solutions are \( x = 16 \) and \( x = 8 \). Rounded to the nearest hundredth, the solutions are the same because they are whole numbers: 16.00 and 8.00.

Step 1: Expression Simplification

The equation is \( (x-12)^{2}=16 \). Start by taking a square root on both sides. Remember, the square root has two values: positive and negative. So, you get \( x-12 = \pm \sqrt{16} \).

Step 2: Extract the Roots

Calculate the square root of 16. It has both a positive and negative solution since \( \sqrt{16} \) is 4 and \( -\sqrt{16} \) is -4. So, the two new equations are: \( x-12 = 4 \) and \( x-12 = -4 \).

Step 3: Solve for x

Finally, solve the two new equations for x. For the first equation, add 12 to both sides which gives \( x = 16 \). For the second equation, add 12 to both sides which gives \( x = 8 \).

Key Concepts

Solving Quadratic Equations

One of the foundational methods for solving quadratic equations is by extracting square roots. Quadratic equations are characterized by the highest power of the variable being squared (for example, the term x^2). To solve such equations, we often manipulate the equation so that a term is squared on one side while a constant remains on the other side, similar to the given exercise, where the equation is simplified to (x - 12)^2 = 16.

Extracting square roots involves applying the square root operation to both sides of the equation. Due to the nature of squaring, we always consider both the positive and negative square roots. This directly leads to finding two solutions, which are frequently called the roots of the equation. As in the provided example, after applying the square root to both sides, we arrive at two separate linear equations, which are then easily solved for the variable x.

Square Root Properties

Understanding square root properties is crucial when extracting square roots to solve equations. A square root of a number x, symbolized as √x, is a value that, when multiplied by itself, gives x. It's important to recall that every positive number has two square roots: one positive and one negative. This is why, in the exercise, we encounter ±√16 leading to ±4, which we then use to find the two potential solutions.

Furthermore, the process of squaring and the operation of square rooting are inverse operations. When you square a square root or vice versa, they cancel each other out, leading to the original number. This relationship is frequently used to isolate the variable in quadratic equations.

Exact vs. Decimal Roots

Students must differentiate between exact roots and decimal approximations. For instance, the square root of 16 is exactly 4, but the square root of 2, for example, would be an endless decimal 1.414..., which we often approximate in our calculations.

Exact and Decimal Solutions

In algebra, we aim to find the most precise answer possible, which are termed 'exact solutions'. These are preferred because they maintain the complete accuracy of the result. For the exercise (x - 12)^2 = 16, the exact solutions are the integers 16 and 8.

However, when exact solutions are not integers or simple fractions, we might settle for decimal solutions which are rounded approximations of the exact values. Decimal solutions are practical, especially when dealing with measurements in real-life situations where absolute precision is not necessary.

In cases where exact roots form complex or infinite decimals, such as with the square root of 2, estimating to a specific decimal place, like to the hundredth 1.41, can be useful. This level of precision often suffices unless a specific degree of accuracy is required, which is typically informed by the context of the problem.