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Chapter 1: Problem 78

Solve each equation in the complex number system. $$ x^{4}+3 x^{2}-4=0 $$

Short Answer

Expert verified
The solutions are \( x = 1 \), \( x = -1 \), \( x = 2i \), and \( x = -2i \).

Step by step solution

01

Introduce a substitution

To simplify the equation, let鈥檚 make a substitution. Letting \( y = x^2 \), the original equation \( x^4 + 3x^2 - 4 = 0 \) transforms into a quadratic equation in terms of \( y \): \[ y^2 + 3y - 4 = 0 \]
02

Solve the quadratic equation

Solve the quadratic equation \( y^2 + 3y - 4 = 0 \). We factorize it: Find two numbers that multiply to -4 and add to 3. These numbers are 4 and -1, so the equation becomes \( (y + 4)(y - 1) = 0 \)
03

Find the solutions for y

Set each factor equal to zero and solve for \( y \): \( y + 4 = 0 \) \( y = -4 \) \( y - 1 = 0 \) \( y = 1 \)
04

Reverse the substitution

Recall the substitution \( y = x^2 \). So we have the equations: \( x^2 = -4 \) \( x^2 = 1 \)
05

Solve for x

Solve each equation for \( x \): For \( x^2 = -4 \), \( x = \pm 2i \) since \( i \) is the imaginary unit where \( i^2 = -1 \). For \( x^2 = 1 \), \( x = \pm 1 \)
06

List all solutions

The solutions for \( x \) are: \( x = 1 \), \( x = -1 \), \( x = 2i \), and \( x = -2i \)

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Key Concepts

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

Quadratic Equations
A quadratic equation is a second-degree polynomial equation in a single variable. It can be written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Quadratic equations appear frequently in algebra and can have two, one, or no real solutions. Sometimes, especially in more complex problems, these solutions turn out to be complex numbers.
Complex Numbers
A complex number is a number that has both a real part and an imaginary part. It is written in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined by the property \( i^2 = -1 \). Complex numbers extend the concept of one-dimensional number lines to two dimensions by combining real and imaginary parts. This is crucial for solving equations like \( x^2 = -4 \), where no real number \( x \) can satisfy the equation. Thus, we use the concept of an imaginary unit \( i \), so \( x = \pm 2i \).
Substitution Method
The substitution method is a mathematical technique used to simplify complex equations by introducing a new variable. In the given problem, solving \( x^4 + 3x^2 - 4 = 0 \) directly can be complicated. By substituting \( y = x^2 \), the equation becomes \( y^2 + 3y - 4 = 0 \), a simpler quadratic equation. After solving for \( y \), reversing the substitution helps to find the original variable \( x \). This method is especially helpful in breaking down higher-degree polynomials into more manageable quadratic equations.
Factorization
Factorization is the process of breaking down a complex expression into a product of simpler factors. For the quadratic equation \( y^2 + 3y - 4 = 0 \), factorizing involves finding two numbers that multiply to \( -4 \) (the constant term) and add to \( 3 \) (the coefficient of the linear term). These numbers are \( 4 \) and \( -1 \): \( (y + 4)(y - 1) = 0 \). We set each factor to zero to find the values of \( y \). Afterward, we reverse the substitution to find the values of \( x \). Essentially, factorization breaks down the quadratic equation into simpler linear factors to find the solutions efficiently.

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Most popular questions from this chapter

Elaine can complete a landscaping project in 2 hours with the help of either her husband Brian or both her two daughters. If Brian and one of his daughters work together, it would take them 4 hours to complete the project. Assuming the rate of work is constant for each person, and the two daughters work at the same rate, how long would it take Elaine, Brian, and one of their daughters to complete the project?

The distance to the surface of the water in a well can sometimes be found by dropping an object into the well and measuring the time elapsed until a sound is heard. If \(t_{1}\) is the time (measured in seconds) that it takes for the object to strike the water, then \(t_{1}\) will obey the equation \(s=16 t_{1}^{2}\), where \(s\) is the distance (measured in feet). It follows that \(t_{1}=\frac{\sqrt{s}}{4}\). Suppose that \(t_{2}\) is the time that it takes for the sound of the impact to reach your ears. Because sound waves are known to travel at a speed of approximately 1100 feet per second, the time \(t_{2}\) to travel the distance \(s\) will be \(t_{2}=\frac{s}{1100} .\) See the illustration. Now \(t_{1}+t_{2}\) is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard. We have the equation $$ \text { Total time elapsed }=\frac{\sqrt{s}}{4}+\frac{s}{1100} $$ Find the distance to the water's surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.

Computing Grades In your Economics 101 class, you have scores of \(68,82,87,\) and 89 on the first four of five tests. To get a grade of B, the average of the first five test scores must be greater than or equal to 80 and less than 90 . (a) Solve an inequality to find the least score you can get on the last test and still earn a \(\mathrm{B}\). (b) What score do you need if the fifth test counts double?

Electricity Rates During summer months in 2018 , Omaha Public Power District charged residential customers a monthly service charge of \(\$ 25,\) plus a usage charge of 10.064 per kilowatt-hour \((\mathrm{kWh})\). If one customer's monthly summer bills ranged from a low of \(\$ 140.69\) to a high of \(\$ 231.23\), over what range did usage vary (in \(\mathrm{kWh}\) )?

A 30.5-ounce can of Hills Bros. \(\begin{array}{llll}\text { coffee } & \text { requires } & 58.9 \pi & \text { square }\end{array}\) inches of aluminum. If its height is 6.4 inches, what is its radius?
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