Problem 56 - Algebra and Trigonometry

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 \(x_1 = -1 + i\) and \(x_2 = -1 - i\).

Step 1: Identify the equation

The given equation is a quadratic equation of the form \[ x^2 + 2x + 2 = 0. \]

Step 2: Recognize the need for the quadratic formula

Since this is a quadratic equation, use the quadratic formula to find the roots. The quadratic formula is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] For our equation, the coefficients are \( a = 1 \), \( b = 2 \), and \( c = 2 \).

Step 3: Calculate the discriminant

The discriminant \( \Delta \) is calculated as \[ \Delta = b^2 - 4ac. \] Substituting the values, we get \[ \Delta = 2^2 - 4 \cdot 1 \cdot 2 = 4 - 8 = -4. \]

Step 4: Apply the quadratic formula

Substitute the values of \( a \), \( b \), and the discriminant into the quadratic formula:\[ x = \frac{-2 \pm \sqrt{-4}}{2 \cdot 1}. \]

Step 5: Simplify the complex roots

Since the discriminant is negative, the roots will be complex. Compute \( \sqrt{-4} = 2i \):\[ x = \frac{-2 \pm 2i}{2}. \]Then simplify the expression:\[ x = -1 \pm i. \]

Step 6: Write the solutions in the form a+bi

The quadratic equation has solutions:\[ x_1 = -1 + i \quad \text{and} \quad x_2 = -1 - i. \] Both solutions are expressed in the form \( a + bi \).

Key Concepts

Complex Numbers

When solving quadratic equations, you may encounter solutions that are not real numbers. These are known as complex numbers. A complex number is expressed in the form \(a + bi\), where:

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

Complex numbers extend the number system so that any polynomial equation has a solution. In the quadratic equation \(x^2 + 2x + 2 = 0\), the roots are complex because the discriminant is negative.

Understanding complex numbers is essential when your standard quadratic formula results in roots involving \(\sqrt{-1}\). This typically occurs when the discriminant is less than zero, indicating all solutions will have imaginary parts. Knowing how to handle such complex solutions is crucial in a variety of mathematical and real-world applications.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is expressed as:

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

This formula provides two solutions, one for the plus sign and one for the minus. It solves the equation by calculating where the parabola, represented by the quadratic equation, crosses the x-axis.

In the case of the equation \(x^2 + 2x + 2 = 0\), the coefficients are \(a = 1\), \(b = 2\), and \(c = 2\). By plugging these values into the quadratic formula, we determine the roots of the equation. This is particularly useful when factoring is difficult or impossible, making the quadratic formula an essential tool in algebra.

Discriminant Calculation

Within the quadratic formula is a key component called the discriminant, represented by \(b^2 - 4ac\). The discriminant helps determine the nature of the roots of a quadratic equation:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root.
• If the discriminant is negative, there are two complex roots.

Calculating the discriminant of \(x^2 + 2x + 2 = 0\), we get \(\Delta = 2^2 - 4 \times 1 \times 2 = -4\). Since \(\Delta\) is negative, this indicates complex roots.

Understanding the discriminant is crucial for predicting the type of solutions without actually solving the equation, providing insight into the polynomial's behavior. This step is important in both solving and understanding the geometry of quadratic equations.