Problem 60 - 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 \( x = -1 + i \) and \( x = -1 - i \).

Step 1: Identify the Coefficients

The given quadratic equation is \( x^2 + 2x + 2 = 0 \). Identify the coefficients: \( a = 1 \), \( b = 2 \), and \( c = 2 \).

Step 2: Use the Quadratic Formula

Recall the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this formula to find the roots of the quadratic equation.

Step 3: Calculate the Discriminant

The discriminant \( \Delta \) is given by \( b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \): \( \Delta = 2^2 - 4 \times 1 \times 2 = 4 - 8 = -4 \).

Step 4: Compute the Square Root of the Discriminant

Since the discriminant \( \Delta = -4 \) is negative, the solutions will involve complex numbers. Compute \( \sqrt{-4} = 2i \).

Step 5: Substitute into the Quadratic Formula

Substitute \( b = 2 \), \( a = 1 \), and \( \sqrt{\Delta} = 2i \) into the quadratic formula:\[ x = \frac{-2 \pm 2i}{2 \times 1} = \frac{-2 \pm 2i}{2} \].

Step 6: Simplify the Expression

Simplify the expression to find the solutions:\[ x = \frac{-2}{2} \pm \frac{2i}{2} = -1 \pm i \].Thus, the solutions are \( x = -1 + i \) and \( x = -1 - i \).

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. It is generally written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In our exercise, the quadratic equation is \( x^2 + 2x + 2 = 0 \). Here:

• \( a = 1 \)
• \( b = 2 \)
• \( c = 2 \)

Quadratic equations can have two solutions, one solution, or no real solution, depending on the discriminant \( \,b^2 - 4ac \). This discriminant gives us crucial information about the nature of the roots, whether real or complex. In the given example, the discriminant helps identify the type of solutions.

Exploring the Quadratic Formula

The quadratic formula is a practical tool to solve any quadratic equation. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this formula, the term \( b \) and \( 4ac \) come from the standard form of the quadratic equation. To find solutions:

• Calculate the discriminant \( b^2 - 4ac \).
• Inspect its sign to determine the nature of solutions: positive for two real solutions, zero for one real solution, and negative for complex solutions.
• Substitute \( a \), \( b \), and \( c \) into the formula and solve for \( x \).

For the equation \( x^2 + 2x + 2 = 0 \), the discriminant is \(-4\). Substituting the values into the formula gives us the solutions, illustrating the power of this formula to solve quadratic equations, even when complex numbers are involved.

Delving Into Complex Solutions

Complex solutions occur when the discriminant of a quadratic equation is negative. Such solutions are made up of a real part and an imaginary part. Imaginary numbers are expressed with the unit \( i \), defined by \( i^2 = -1 \).

In our problem, the discriminant is \(-4\), which translates into an imaginary number when taking the square root. We find \( \sqrt{-4} = 2i \). Thus, the solutions are complex:

• For \( x = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \), we get \(-1 + i \).
• For \( x = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \), we get \(-1 - i \).

These complex numbers indicate the direction or rotation in the complex plane, each comprising a real and an imaginary component. Understanding complex solutions is crucial, as they reveal more about the nature of quadratic equations beyond what real numbers can show.