Problem 18 - College Algebra with Modeling and Visualization

Question

Exercises \(1-28:\) Solve the quadratic equation. Check your answers for Exercises \(1-12\). $$ -3(x+5)^{2}-6=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -5 + i\sqrt{2}\) and \(x = -5 - i\sqrt{2}\).

Step 1: Simplify the Equation

First, we start by moving all terms involving variables to one side of the equation for simplicity. Subtracting 6 from both sides gives:\[-3(x+5)^2 = 6\]

Step 2: Isolate the Quadratic Term

Next, divide both sides of the equation by -3 to isolate the quadratic term:\[(x+5)^2 = -2\]

Step 3: Take the Square Root of Both Sides

To solve for \(x\), take the square root of both sides of the equation. Remember to consider both the positive and negative roots:\[x+5 = \pm \sqrt{-2}\]This indicates imaginary solutions since the square root of a negative number is imaginary.

Step 4: Solve for x

Subtract 5 from both sides to solve for \(x\):\[x = -5 \pm i\sqrt{2}\]This provides two solutions, which are imaginary:

Step 5: Conclusion

After all steps, we find the solutions \(x = -5 + i\sqrt{2}\) and \(x = -5 - i\sqrt{2}\). Both solutions are complex numbers due to the presence of the imaginary unit \(i\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and come in the form of \(ax^2 + bx + c = 0\). These equations involve the variable \(x\) raised to the second power. The standard form includes:

• \(a\): The coefficient of \(x^2\)
• \(b\): The coefficient of \(x\)
• \(c\): The constant term

A quadratic equation can result in different types of solutions. These include one real solution, two real solutions, or two complex solutions. Solving these equations may require various methods, such as factoring, using the quadratic formula, or completing the square. Understanding the nature of solutions is crucial, especially when they are not real numbers.
Quadratic equations often appear in problems involving projectile motion, calculating areas, and more. Recognizing the structure and solving these equations is an essential skill for understanding mathematics more deeply.

Imaginary Solutions

An interesting aspect of quadratic equations is the potential for imaginary solutions. Imaginary numbers arise when we take the square root of a negative number. They are part of the broader family of complex numbers, expressed as \(a + bi\), where:

• \(a\) is the real part
• \(b\) is the imaginary part
• \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\)

In the provided exercise, taking the square root of -2 leads us to \(\pm i\sqrt{2}\). Here, \(i\) indicates the imaginary nature. Imaginary solutions often occur in quadratic equations without real roots. They reveal how extensive the number system is, bridging gaps when real numbers fall short.
Imaginary numbers, although initially challenging, have practical applications in various fields such as engineering, physics, and signal processing. Understanding complex numbers enhances problem-solving and mathematical modeling capabilities.

Completing the Square

Completing the square is a valuable algebraic technique used to solve quadratic equations. It involves transforming a quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. This method works well when factoring is difficult or impossible.
To complete the square, follow these general steps:

• Ensure the coefficient of \(x^2\) is 1 (if not, factor it out)
• Take half of the \(x\) coefficient (\(b\)), square it, and add it to both sides of the equation
• Rearrange the equation into a perfect square form \((x + d)^2\)
• Solve for \(x\) using the square root method

In the original step-by-step solution, a similar approach was applied indirectly by isolating \((x+5)^2\) and then addressing imaginary solutions. Completing the square provides a systematic method to reach the solution, offering clarity and ease to otherwise complex equations.
Practicing this method can deepen understanding of equations and improve algebraic manipulation skills, preparing students for advanced mathematical challenges.