/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Solve each equation by the squar... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 25

Solve each equation by the square root property. $$x^{2}=16$$

Short Answer

Expert verified
x = 4, -4

Step by step solution

01

- Understand the Square Root Property

The square root property states that if \(x^2 = k\), then \(x = \pm \sqrt{k}\). This means we need to take the square root of both sides of the equation \(x^2 = 16\).
02

- Apply the Square Root Property

Taking the square root of both sides of \(x^2 = 16\) gives \(x = \pm \sqrt{16}\).
03

- Evaluate the Square Root

Calculate \sqrt{16}. Since \sqrt{16} = 4, the equation becomes \(x = \pm 4\).
04

- Write the Solution

Express the final solution. Thus, the solutions are \(x = 4\) and \(x = -4\).

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Key Concepts

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

solving quadratic equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They can be solved in several ways, such as factoring, completing the square, using the quadratic formula, or by employing the square root property. For our specific example, we used the square root property. This technique is particularly fast and efficient when the quadratic equation has no linear term (the term with \(bx\) is missing) and can be easily transformed into the form \(x^2 = k\). In such cases, we can isolate the \(x^2\) term and then use the property to solve for \(x\). This approach simplifies the process and quickly leads to the correct solutions.
square roots
A square root is a number which, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because \(4 \times 4 = 16\). The process of finding the square root is denoted by the symbol \(\sqrt{}\). It is important to remember that square roots can be both positive and negative. This is because both \(4 \times 4 = 16\) and \(-4 \times -4 = 16\). Thus, \(\sqrt{16} = 4\) and \(\sqrt{16} = -4\). Always keep in mind this dual nature when solving equations that involve square roots.
positive and negative solutions
When you solve an equation like \(x^2 = 16\) using the square root property, you must include both the positive and negative solutions. This is because squaring either a positive number or a negative number will result in a positive value. So, for \(x^2 = 16\), taking the square root of both sides yields \(x = \pm 4\), giving us two solutions: \(x = 4\) and \(x = -4\). It's essential to write both solutions to fully solve the equation. To summarize, always consider both the positive and negative square roots when dealing with quadratic equations and square roots.

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Most popular questions from this chapter

If p units of an item are sold for \(x\) dollars per unit, the revenue is \(R=p x\). Use this idea to analyze the following problem. Number of Apartments Rented The manager of an 80-unit apartment complex knows from experience that at a rent of \(\$ 300,\) all the units will be full. On the average, one additional unit will remain vacant for each \(\$ 20\) increase in rent over \(\$ 300 .\) Furthermore, the manager must keep at least 30 units rented due to other financial considerations. Currently, the revenue from the complex is \(\$ 35,000 .\) How many apartments are rented? According to the problem, the revenue currently generated is \(\$ 35,000 .\) Substitute this value for revenue into the equation from Exercise \(53 .\) Solve for \(x\) to answer the question in the problem.

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Complex numbers are used to describe current I, voltage \(E,\) and impedance \(Z\) (the opposition to current). These three quantities are related by the equation \(E=I Z, \quad\) which is known as Ohm's Law. Thus, if any two of these quantities are known, the third can be found. In each exercise, solve the equation \(E=I Z\) for the remaining value. $$I=20+12 i, Z=10-5 i$$

Solve each rational inequality. Write each solution set in interval notation. $$\frac{-6}{3 x-5} \leq 2$$
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