Problem 29 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by extracting square roots. List both the exact answer and a decimal answer that has been rounded to two decimal places. $$ (x-12)^{2}=18 $$

Step-by-Step Solution

Answer

Exact solutions are \(x = 12 + \sqrt{18}\) and \(x = 12 - \sqrt{18}\). The rounded decimal solutions for x to two decimal places are 16.24 and 7.76

Step 1: Square rooting both sides of the equation

The square root of \((x-12)^{2}\) is \(x-12\) and the square root of 18 is \(\pm\sqrt{18}\). This results in the equation \(x - 12 = \pm\sqrt{18}\)

Step 2: Solving for x

Solving for x leads to two equations: \(x = 12 + \sqrt{18}\) and \(x = 12 - \sqrt{18}\)

Step 3: Finding the decimal rounding of x

The decimal rounding leads to two possible values for x. Approximating \(12 + \sqrt{18}\) to the nearest two decimal places gives 12 + 4.24 = 16.24 and approximating \(12 - \sqrt{18}\) to the nearest two decimal places gives 12 - 4.24 = 7.76.

Key Concepts

Extracting Square Roots

Solving a quadratic equation by extracting square roots is a method that can be applied when the equation is in the form of (variable)^2 = number. This approach is often seen as more straightforward than other methods as it involves fewer steps and directly utilizes the property that a square root and a square are inverse operations.

Consider the equation \( (x-12)^2=18 \). Here, we isolate the squared term before taking the square root of both sides. Doing so results in eliminating the square on the left, leaving us with \( x-12 \) and \( \pm\sqrt{18} \) on the right. The ± sign indicates that there are two possible solutions to the equation, because squaring either a positive or negative value results in a positive result.

It's important to emphasize embracing both solutions, as neglecting either the positive or negative root can lead to an incomplete answer set. Remember, whenever you introduce a radical, consider both its positive and negative values.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation, regardless of how it is arranged. When an equation cannot be easily simplified for the application of square roots, the quadratic formula comes into play. The formula is \( x= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), where \( a \), \( b \) and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).

This method guarantees you will find the solutions to the equation, if they exist, without the need to rearrange the equation significantly. The formula takes into account the possibilities of both real and complex roots, indicated by the discriminant \( b^2-4ac \). If this part of the formula is positive, there are two real solutions. If it is zero, there is one real solution. If it's negative, the solutions are complex. Being comfortable with the quadratic formula allows you to confidently tackle any quadratic equation you encounter.

Decimal Approximation

Decimal approximation involves converting an exact number, particularly an irrational number like \( \sqrt{18} \), into a decimal number up to a certain degree of accuracy. In the context of solving quadratic equations, this is often necessary when providing an answer that is easier to comprehend or when a simplified numeric answer is required.

Using \( \sqrt{18} \), which simplifies to approximately 4.24264, you might round to the nearest hundredths place, giving you 4.24. In the equation \( x = 12 + \sqrt{18} \) and \( x = 12 - \sqrt{18} \) from the exercise, this rounding leads to the approximate solutions 16.24 and 7.76, respectively. When rounding, it's crucial to remember that small errors can be introduced, but for many practical purposes, these approximations are sufficient and necessary, especially when exact answers are difficult to use, such as in measurements or real-world problem-solving scenarios.

Taking decimal approximation into consideration when presenting the final solutions forms a bridge between the exact mathematical world and its practical applications in everyday life. It's an indispensable skill for a student to learn.