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Chapter 0: Problem 77

Solve the equation by extracting square roots. $$ (x+2)^{2}=14 $$

Short Answer

Expert verified
The solutions for the equation \((x+2)^2 = 14\) are \(x = 1.74\) and \(x = -5.74\).

Step by step solution

01

Take the Square Root

The first step is to take the square root of each side of the equation. This yields two possible solutions, as the square root of any number can be either positive or negative. So, the equation becomes: \(x+2 = \pm \sqrt{14}\).
02

Simplify the Equation

The square root of 14, to two decimal places, is 3.74, this gives: \(x+2 = \pm 3.74\).
03

Solve for x

Next, solve for x in the equation \(x+2 = 3.74\) and \(x+2 = -3.74\). The solutions will be: \(x = 3.74 - 2 = 1.74\) and \(x = -3.74 - 2 = -5.74\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Method
The square root method is a helpful technique used to solve quadratic equations of the form \((x + a)^2 = b\). In such equations, our goal is to isolate the variable to find its value. Here's how it works:
  • First, take the square root of both sides of the equation. This helps you simplify the equation to its linear form. Remember, when you take the square root, you must consider both the positive and negative roots. Thus, you get two potential solutions.
  • In our original problem, we started with \((x+2)^2 = 14\). By taking the square root, the equation transforms to \(x + 2 = \pm \sqrt{14}\).
This method is particularly useful when dealing with simple quadratic equations where isolating the squared term is straightforward. It鈥檚 important to note that not all quadratic equations can be solved using this method, especially if they have more complex structures.
Extracting Square Roots
Extracting square roots is a crucial step in solving equations like our exercise. When you encounter a squared term isolated on one side, square roots allow you to find the possible values of the variable:
  • Identify the squared term and move other terms if needed to isolate it on one side.
  • Apply the square root to both sides. This introduces the 卤 sign, accounting for both positive and negative root possibilities.
  • Further simplify the equation by calculating the decimal value of the square root, if necessary, which makes the equation easier to handle.
In our example, simplifying \(\sqrt{14}\) gives approximately 3.74. This leads to two equations: \(x + 2 = 3.74\) and \(x + 2 = -3.74\). Solving these will provide the two possible solutions for \(x\). Understanding how to properly extract square roots is fundamental in algebra, preparing you for more advanced topics.
Trigonometry Concepts
While the original exercise doesn't directly involve trigonometry, understanding the importance of extracting square roots is a skill also useful in trigonometry. Many trigonometric equations can lead to expressions involving squares, making this technique essential:
  • For example, in trigonometry, the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) may require solving for \(\sin \theta\) or \(\cos \theta\) by using square roots.
  • Similarly, expressions like \((\tan \theta)^2\) might need square root extraction to find the angle, which is often part of analyzing wave functions or optimizing angles in physics problems.
Thus, mastering extracting square roots isn't just for solving polynomial equations鈥攊t enhances your skills in handling various mathematical challenges, including those in trigonometry.

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Most popular questions from this chapter

Geometry A rectangle is bounded by the \(x\)-axis and the semicircle \(y=\sqrt{36-x^{2}}\) (see figure). Write the area \(A\) of the rectangle as a function of \(x\), and determine the domain of the function.

Temperature The table shows the temperature \(y\) (in degrees Fahrenheit) of a certain city over a 24-hour period. Let \(x\) represent the time of day, where \(x=0\) corresponds to \(6 \mathrm{~A}\).M. $$ \begin{array}{|c|c|} \hline \text { Time, } \boldsymbol{x} & \text { Temperature, } \boldsymbol{y} \\\ \hline 0 & 34 \\ 2 & 50 \\ 4 & 60 \\ 6 & 64 \\ 8 & 63 \\ 10 & 59 \\ 12 & 53 \\ 14 & 46 \\ 16 & 40 \\ 18 & 36 \\ 20 & 34 \\ 22 & 37 \\ 24 & 45 \\ \hline \end{array} $$ A model that represents these data is given by \(y=0.026 x^{3}-1.03 x^{2}+10.2 x+34, \quad 0 \leq x \leq 24 .\) (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the graph to approximate the times when the temperature was increasing and decreasing. (d) Use the graph to approximate the maximum and minimum temperatures during this 24 -hour period. (e) Could this model be used to predict the temperature for the city during the next 24 -hour period? Why or why not?

Think About It Consider \(f(x)=\sqrt{x-2}\) and \(g(x)=\sqrt[3]{x-2}\). Why are the domains of \(f\) and \(g\) different?

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from a height of \(6.5\) feet at a velocity of 72 feet per second. $$ t_{1}=0, t_{2}=4 $$

True or False? Determine whether the statement is true or false. Justify your answer. If \(f\) is an even function, determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=f(x-2)\)
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