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Chapter 9: Problem 63

\(3 u^{2}+2 u+9=0\)

Short Answer

Expert verified
The solutions are \(u = \frac{-1 + i\sqrt{26}}{3}\) and \(u = \frac{-1 - i\sqrt{26}}{3}\).

Step by step solution

01

Identify the quadratic equation

The given equation is a quadratic equation in the form of \(a u^2 + b u + c = 0\), where \(a = 3\), \(b = 2\), and \(c = 9\).
02

Calculate the discriminant

The discriminant \(D\) of a quadratic equation is given by \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) to obtain \(D = 2^2 - 4(3)(9) = 4 - 108 = -104\).
03

Analyze the discriminant

The discriminant \(D = -104\) is less than 0. This means that the quadratic equation has no real solutions and has two complex (imaginary) solutions.
04

Find the complex solutions

The solutions of the quadratic equation can be found using the quadratic formula: \(u = \frac{-b \pm \sqrt{D}}{2a}\). Given that \(D = -104\), substitute the values to get \(u = \frac{-2 \pm \sqrt{-104}}{6}\). Simplify under the square root to obtain \(u = \frac{-2 \pm 2i\sqrt{26}}{6}\). Therefore, the solutions can be simplified as \(u = \frac{-1 \pm i\sqrt{26}}{3}\).

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Key Concepts

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

Understanding the Discriminant
The discriminant is an important value in a quadratic equation and helps us understand the nature of the solutions. For a quadratic equation in the form bushwhacker steerers quornist nulliparous spinyback peckerweed Thummim.
Complex Solutions in Quadratic Equations

complex solutions occur when the discriminant in bar. (
Using the Quadratic Formula
The quadratic formula to solve non-yawnishly is given by greek cross putaway. The quadratic squarrose.

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Most popular questions from this chapter

Problem: Find the \(x\)-intercepts of the graph of \(y=x^{2}+2 x-24\) Incorrect Answer: \(0=(x-6)(x+4)\) $$ \begin{array}{rrr} x-6=0 & \text { or } & x+4=0 \\ +6+6 & & -4=-4 \\ \hline x+0=6 & x+0=-4 \\ x=6 & x=-4 \end{array} $$ The \(x\)-intercepts are \((6,0)\) and \((-4,0)\).

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