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Chapter 2: Problem 8

Find real numbers \(a\) and \(b\) such that the equation is true. $$a+b i=13+4 i$$

Short Answer

Expert verified
The real numbers \(a\) and \(b\) such that the equation \(a+b i=13+4 i\) is true are \(a=13\) and \(b=4\).

Step by step solution

01

Identify the Real and Imaginary Parts

Consider the equation \(a+b i=13+4 i\). Here \(a+b i\) and \(13+4 i\) are complex numbers. The real part of a complex number is the part without \(i\), and the imaginary part is the part with \(i\). So, in this equation, \(a\) and \(13\) are the real parts, while \(b i\) and \(4 i\) are the imaginary parts.
02

Equate the Real Parts and Solve

From the equation, the real part on the left side should be equal to the real part on the right side. This gives us the equation \(a=13\). Here no calculation is necessary because the equation is already solved. So, \(a=13\). This is the first part of the answer.
03

Equate the Imaginary Parts and Solve

Now, equate the imaginary parts from both sides. This gives us \(b i = 4 i\). The \(i\) in both sides can be cancelled out, leaving \(b = 4\). So, we have found the value of \(b\), which is the second part of the answer.

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Most popular questions from this chapter

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$h(x)=2 x^{4}-3 x+2$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=x^{3}+24 x^{2}+214 x+740$$

The maximum safe load uniformly distributed over a one-foot section of a two- inch-wide wooden beam can be approximated by the model $$\text { Load }=168.5 d^{2}-472.1$$ where \(d\) is the depth of the beam. (a) Evaluate the model for \(d=4, d=6, d=8, d=10\) and \(d=12 .\) Use the results to create a bar graph. (b) Determine the minimum depth of the beam that will safely support a load of 2000 pounds.

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x-4=0$$
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