Problem 60 - Precalculus Mathematics for Calculus

Question

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}+2 x+2=0$$

Step-by-Step Solution

Answer

The solutions are \(-1+i\) and \(-1-i\).

Step 1: Identify the Quadratic Equation

We need to solve the quadratic equation given by \(x^{2}+2x+2=0\). This is a standard quadratic equation, so we'll use the quadratic formula to find the solutions.

Step 2: Apply the Quadratic Formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a=1\), \(b=2\), and \(c=2\). Substituting these values into the quadratic formula gives:\[x=\frac{-2\pm\sqrt{2^{2}-4\cdot1\cdot2}}{2\cdot1}\] Simplifying inside the square root, we get \(x=\frac{-2\pm\sqrt{4-8}}{2}\).

Step 3: Simplify the Expression Under the Square Root

Now, we simplify the expression under the square root: \(4-8=-4\). So the equation becomes:\[x=\frac{-2\pm\sqrt{-4}}{2}\] We know that \(\sqrt{-4}=2i\), therefore:\[x=\frac{-2\pm2i}{2}\]

Step 4: Find the Two Solutions

Split the expression to find the two potential solutions:1. \(x=\frac{-2+2i}{2}\)2. \(x=\frac{-2-2i}{2}\)Simplifying each of these gives:\(x=-1+i\) and \(x=-1-i\).

Step 5: Conclusion and Check Solutions

Therefore, the solutions to the equation \(x^{2}+2x+2=0\) are \(-1+i\) and \(-1-i\). We have expressed the solutions in the form \(a+bi\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). This formula provides a straightforward method to find the roots of any quadratic equation, whether the solutions are real or complex.
This is the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

To use the formula, you need to identify the coefficients \(a\), \(b\), and \(c\) in your equation. Let's break down the steps:

• Identify \(a\), \(b\), and \(c\) from the equation.
• Substitute these values into the formula.
• Calculate the discriminant \(b^2 - 4ac\).
• Evaluate \(\sqrt{b^2 - 4ac}\).
• Simplify to find the values of \(x\).

The quadratic formula works beautifully with any quadratic equation, giving you a reliable solution process every time.

Imaginary Numbers

Imaginary numbers appear when we take the square root of a negative number. These numbers are based on the imaginary unit \(i\), where \(i = \sqrt{-1}\). This concept extends the real number system to the complex number system, allowing for solutions to equations that have no real solutions.
In the quadratic formula, when the discriminant (\(b^2 - 4ac\)) is negative, the square root of this value will be an imaginary number.
For example, if the discriminant is \(-4\), then \(\sqrt{-4} = 2i\) because:

• \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1}\)
• \(= 2 \cdot i = 2i\)

Using imaginary numbers, we can express solutions of the form \(a + bi\), where \(a\) and \(b\) are real numbers. This means we have a standard way to describe numbers beyond the real line, enhancing our understanding and providing solutions to previously unsolvable problems.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. There are various methods to solve quadratic equations, but the quadratic formula is one of the most universally applicable.
Here's how you can solve a quadratic equation using the quadratic formula:

• Set the equation to zero, ensuring it follows the format \(ax^2 + bx + c = 0\).
• Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), to find the solutions.
• If the discriminant \(b^2 - 4ac\) is positive, you'll get two real solutions. If it is zero, there's exactly one real solution.
• If the discriminant is negative, the solutions are complex or imaginary, and will be expressed in the form \(a + bi\).

In our original problem, the quadratic formula shows us both solutions. By substituting and simplifying, we find our solutions are \(-1 + i\) and \(-1 - i\). This process not only finds the solutions but also ensures they are expressed in a standard format that includes imaginary components when necessary.