Problem 63 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(c = 9\).

Step 2: 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 \(D = 2^2 - 4(3)(9) = 4 - 108 = -104\).

Step 3: 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.

Step 4: 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 \(u = \frac{-1 \pm i\sqrt{26}}{3}\).

Key Concepts

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.