/*! 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 58 Find the nonreal complex solutio... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 58

Find the nonreal complex solutions of each equation. \(4 x^{2}+5 x+5=0\)

Short Answer

Expert verified
The nonreal complex solutions are \( \frac{-5 + i \sqrt{55}}{8} \) and \( \frac{-5 - i \sqrt{55}}{8} \).

Step by step solution

01

Identify coefficients

The given quadratic equation is in the form of \(ax^2 + bx + c = 0\). Here, \(a = 4\), \(b = 5\), and \(c = 5\).
02

Calculate the discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\): \[D = 5^2 - 4(4)(5) = 25 - 80 = -55\]. Since \(D < 0\), the quadratic equation has nonreal (complex) solutions.
03

Use the quadratic formula

The quadratic formula to find the roots of \(ax^2 + bx + c = 0\) is \[x = \frac{-b \pm \sqrt{D}}{2a}\]. Since \(D = -55\), which is negative, we need to use imaginary numbers. Hence, \[ \sqrt{-55} = i \sqrt{55}\]:\[ x = \frac{-5 \pm i \sqrt{55}}{8} \].
04

Simplify the solutions

So, the solutions are: \[ x = \frac{-5 + i \sqrt{55}}{8}\] and \[x = \frac{-5 - i \sqrt{55}}{8}\].

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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 any equation that can be rearranged into the form \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants. The highest power of the variable \(x\) is 2. Quadratic equations often have two solutions, which may be real or complex. These equations form a parabola when graphed on a coordinate plane. They are used in physics, engineering, and many real-world applications.
discriminant
The discriminant is a key concept in determining the nature of the solutions for a quadratic equation. It is calculated using the formula: \[ D = b^2 - 4ac \] The value of \(D\) tells us the nature of the roots:
  • If \(D > 0\), the equation has two distinct real solutions.
  • If \(D = 0\), the equation has one real solution (a repeated root).
  • If \(D < 0\), the equation has two complex solutions.
In the given example, since \(D = -55\), we have two nonreal (complex) solutions.
quadratic formula
The quadratic formula is used to find the solutions of any quadratic equation. It is expressed as: \[ x = \frac{-b \, \pm \, \sqrt{D}}{2a} \] where \(D\) is the discriminant. By substituting the values of \(a\), \(b\), and \(c\) into this formula, we can find the roots of the equation. For our exercise, \(a = 4\), \(b = 5\), and \(c = 5\), so \[ x = \frac{-5 \, \pm \, \sqrt{-55}}{8} \] This will give us the two roots of the equation.
imaginary numbers
Imaginary numbers are used to extend the real number system. They are based on the imaginary unit \(i\) where \(i^2 = -1\). This is crucial for solving quadratic equations where the discriminant is negative. For example, \( \sqrt{-55} = i \sqrt{55} \) This helps in finding complex solutions for a quadratic equation. In our exercise, the solutions become: \[ x = \frac{-5 \, + \, i \sqrt{55}}{8} \] and \[ x = \frac{-5 \, - \, i \sqrt{55}}{8} \] which are the nonreal complex solutions. By understanding imaginary numbers, we can work with and solve equations that involve such numbers.

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Most popular questions from this chapter

Use the discriminant to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved by factoring or whether the quadratic formula should be used. Do not actually solve. $$ 3 x^{2}=5 x+2 $$

Solve each problem. When appropriate, round answers to the nearest tenth. A 13 -ft ladder is leaning against a house. The distance from the bottom of the ladder to the house is 7 ft less than the distance from the top of the ladder to the ground. How far is the bottom of the ladder from the house?

Solve each equation by completing the square. Give (a) exact solutions and (b) solutions rounded to the nearest thousandth. \((x+1)(x+3)=2\)

Solve each equation. $$ \sqrt{2 x+6}=x-1 $$

Find the nonreal complex solutions of each equation. \(3 r^{2}+4 r+4=0\)
See all solutions

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