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Chapter 9: Problem 77

Find the nonreal complex solutions of each equation. \((r-5)^{2}=-4\)

Short Answer

Expert verified
5 + 2i and 5 - 2i

Step by step solution

01

- Isolate the squared term

Start by isolating the squared term on one side of the equation. The given equation is \[(r-5)^{2} = -4\].
02

- Take the square root of both sides

Apply the square root to both sides of the equation to solve for \(r-5\). Remember to include both the positive and negative roots: \(\sqrt{(r - 5)^2} = \pm \sqrt{-4}\). This simplifies to \(r-5 = \pm 2i\).
03

- Solve for r

Solve for \r\ by isolating it in the equation \(r - 5 = \pm 2i\). This gives: \(r = 5 \pm 2i\).

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Key Concepts

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

Nonreal Numbers
Nonreal numbers are numbers that cannot be placed on the real number line. They include complex numbers, which consist of a real part and an imaginary part. The imaginary unit is denoted by 'i' and is defined as the square root of -1. For example, the nonreal number 3 + 2i has a real part of 3 and an imaginary part of 2i.
Understanding nonreal numbers is crucial for solving equations that do not have real solutions. These numbers often emerge when dealing with the square roots of negative numbers.
Quadratic Equations
A quadratic equation is a second-order polynomial equation in a single variable. It has the general form:
ax^2 + bx + c = 0
In our example, (r-5)虏 = -4 is a quadratic equation since it involves a square term. To solve quadratic equations, we can use methods such as factoring, completing the square, and the quadratic formula. In this particular case, we isolate the squared term and take the square root of both sides. Quadratic equations often reveal complex solutions, especially when the discriminant (b虏 - 4ac) is negative.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 * 3 = 9. However, the square root of a negative number introduces imaginary or complex numbers. In our example, the square root of -4 is 
卤 2i
because (2i)虏 = -4 and (-2i)虏 = -4.

To solve equations like 
(r - 5)^2 = -4
first, we take the square root of both sides, remembering to include both the positive and negative roots. This simplifies to: 
r - 5 = 卤 2i
Finally, we solve for r to get our complex solutions. Understanding how to handle square roots is essential for solving quadratic equations and finding nonreal solutions.

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Most popular questions from this chapter

Professor Barbu has found that the number of students attending his intermediate algebra class is approximated by $$ S(x)=-x^{2}+20 x+80 $$ where \(x\) is the number of hours that the Campus Center is open daily. Find the number of hours that the center should be open so that the number of students attending class is a maximum. What is this maximum number of students?

Graph each parabola. Plot at least two points as well as the vertex. Give the vertex, axis, domain, and range . $$ f(x)=(x+2)^{2}-1 $$

The equations in Exercises \(69-82\) are not grouped by type. Solve equation. Exercises 81 and 82 require knowledge of complex numbers. See Examples I and \(4-7\). $$ 8 x^{6}+513 x^{3}+64=0 $$

Graph each parabola. Plot at least two points as well as the vertex. Give the vertex, axis, domain, and range . $$ f(x)=-\frac{1}{2}(x+1)^{2}+2 $$

Graph each parabola. Plot at least two points as well as the vertex. Give the vertex, axis, domain, and range . $$ f(x)=-x^{2}+2 $$
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