/*! 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 69 Solve each equation in the compl... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 69

Solve each equation in the complex number system. $$ 5 x^{2}+1=2 x $$

Short Answer

Expert verified
The solutions are: \(x = \frac{1}{5} + \frac{2i}{5}\) and \(x = \frac{1}{5} - \frac{2i}{5}\).

Step by step solution

01

Bring the equation to standard quadratic form

Start by rearranging the equation so that all terms are on one side. Subtract 2x from both sides to get:\[5x^2 - 2x + 1 = 0\]
02

Identify coefficients for the quadratic formula

In the standard quadratic form \(ax^2 + bx + c = 0\), identify the values of \(a\), \(b\), and \(c\):a = 5b = -2c = 1
03

Use the quadratic formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with the identified coefficients:\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 5 \cdot 1}}{2 \cdot 5}\] Simplify it to: \[x = \frac{2 \pm \sqrt{4 - 20}}{10}\]
04

Simplify under the square root

Simplify the expression inside the square root:\[-16 = -1 \cdot 16\] So, the expression becomes: \[x = \frac{2 \pm \sqrt{4 - 20}}{10} = \frac{2 \pm \sqrt{-16}}{10}\]This means the value under the square root is complex: \(\sqrt{-16} = 4i\)
05

Simplify the expression

Now substitute back the value of the square root:\[x = \frac{2 \pm 4i}{10}\] Separate into two expressions:\[x = \frac{2}{10} + \frac{4i}{10} = \frac{1}{5} + \frac{2i}{5}\] and \[x = \frac{2}{10} - \frac{4i}{10} = \frac{1}{5} - \frac{2i}{5}\]
06

Write the final solutions

The solutions are:\[ x = \frac{1}{5} + \frac{2i}{5} \] and \[ x = \frac{1}{5} - \frac{2i}{5} \]

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Key Concepts

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

quadratic formula
The quadratic formula is an essential tool used for solving quadratic equations. The formula is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula helps you find the roots of any quadratic equation of the format \( ax^2 + bx + c = 0 \).

Here鈥檚 how you can use the quadratic formula effectively:
  • Identify coefficients: Determine the values of \(a\), \(b\), and \(c\) from the quadratic equation.
  • Apply the formula: Substitute these values into the quadratic formula.
  • Solve for x: Perform the arithmetic operations to find the values of \(x\).

In our exercise, we identified \(a = 5\), \(b = -2\), and \(c = 1\).

By substituting these values into the quadratic formula, we started to solve for the roots of the equation.
complex numbers
Complex numbers arise when solving quadratic equations, especially when the value under the square root (called the discriminant) is negative.

A complex number is written in the form \( a + bi \), where:
  • \

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Most popular questions from this chapter

Challenge Problem Show that the real solutions of the equation \(a x^{2}+b x+c=0, a \neq 0,\) are the reciprocals of the real solutions of the equation \(c x^{2}+b x+a=0\), \(c \neq 0 .\) Assume that \(b^{2}-4 a c \geq 0\)

Describe three ways that you might solve a quadratic equation. State your preferred method; explain why you chose it.

Challenge Problem Solve: \(x-4<2 x-3 \leq \frac{x+5}{3}\)

Field Design A football field, 50 feet wide, is sloped from the center toward the sides for drainage. The height \(h\) in feet, of the field, \(x\) feet from the side, is given by \(h=-0.00025 x^{2}+0.04 x .\) Find the distance from the side when the height is 1.1 feet. Round to the nearest tenth of a foot.

In going from Chicago to Atlanta, a car averages 45 miles per hour, and in going from Atlanta to Miami, it averages 55 miles per hour. If Atlanta is halfway between Chicago and Miami, what is the average speed from Chicago to Miami? Discuss an intuitive solution. Write a paragraph defending your intuitive solution. Then solve the problem algebraically. Is your intuitive solution the same as the algebraic one? If not, find the flaw.
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