Problem 26 - Introductory Algebra for College Students

Question

Solve each quadratic equation using the quadratic formula. $$x^{2}+2 x+2=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(x^{2} + 2x + 2 = 0\) are \(x = -1 + i\) and \(x = -1 - i\).

Step 1: Identify the coefficients

In the equation \(x^{2} + 2x + 2 = 0\), the coefficient \(a = 1\), \(b = 2\), and \(c = 2\)

Step 2: Substitute the coefficients into the quadratic formula

The quadratic formula is: \(x = {-b \pm \sqrt{b^{2} - 4ac} \over 2a}\). Substitute the coefficients into the formula to get: \(x = {-2 \pm \sqrt{(2)^{2} - 4(1)(2)} \over 2(1)}\) = \({-2 \pm \sqrt{4-8} \over 2}\)

Step 3: Simplify the result

Simplify inside the square root: \(x = {-2 \pm \sqrt{-4} \over 2}\). As the result inside the square root is a negative number, this results in an imaginary number. The solution is: \(x = {-2 \pm 2i \over 2}\) = \({-1 \pm i}\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). They are fundamental in algebra and appear frequently in math problems. The characteristic feature of a quadratic equation is its second-degree term, \( ax^2 \), which ensures that the equation will graph as a parabola when plotted. The values of \( a \), \( b \), and \( c \) are called coefficients and can be any real or complex number.

Solving quadratic equations can be approached in different ways:

• Factoring: Finding two numbers that multiply to \( ac \) and add to \( b \).
• Completing the square: Rewriting the equation to make it easier to solve.
• Quadratic formula: The most general method, applicable to any quadratic equation:
  \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In the quadratic formula, \( b^2 - 4ac \) is known as the discriminant. It helps determine the nature of the solutions:

• If positive, there are two real solutions.
• If zero, there is one real solution.
• If negative, solutions are complex or involve imaginary numbers.

Imaginary Numbers

Imaginary numbers arise when we solve equations that involve the square root of negative numbers. In regular mathematics, negative numbers don’t have real square roots. This is where imaginary numbers come in, making it possible to work with square roots of negative values.

The basic unit of imaginary numbers is \( i \), defined as the square root of \( -1 \). Thus, \( i^2 = -1 \). When we encounter an equation like \( \sqrt{-4} \), we express it using imaginary numbers:
\[ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \]
Imaginary numbers, when combined with real numbers, form complex numbers, usually written in the form:
\( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

In solving the quadratic equation \( x^2 + 2x + 2 = 0 \), the discriminant \( b^2 - 4ac = -4 \) indicates the solutions contain imaginary numbers. This gives our solutions as \( x = -1 \pm i \). These solutions are complex and occur as conjugates, reflecting the symmetry in how complex numbers behave.

Coefficients in Algebra

Coefficients are the numerical or constant parts of terms in a polynomial expression. They play an essential role in determining the behavior and solutions of algebraic equations. In our equation \( x^2 + 2x + 2 = 0 \), the coefficients are \( a = 1 \), \( b = 2 \), and \( c = 2 \).

Understanding coefficients helps you to:

• Identify the equation type: The highest power of \( x \) in the terms shows the degree of the equation.
• Apply the quadratic formula: Use coefficients in \( ax^2 + bx + c = 0 \) directly in the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Analyze solutions: Coefficients affect the nature of solutions through the discriminant \( b^2 - 4ac \).

By substituting \( a \), \( b \), and \( c \) into the quadratic formula, we solve for \( x \). Each coefficient has a specific role, with \( a \) affecting the concavity of a parabola, \( b \) its symmetry, and \( c \) as the constant term impacting the vertical shift. Together, they guide the path to finding the roots of the equation.