Problem 118 - Intermediate Algebra

Question

Simplify. Determine whether \(-1+i\) is a solution of \(x^{2}+2 x=-2\)

Step-by-Step Solution

Answer

Yes, \(-1+i\) is a solution of \(x^2 + 2x = -2\).

Step 1: Substitute into the Equation

We need to check if \(-1+i\) is a solution to the given equation. To do this, we substitute \(-1+i\) for \(x\) in \(x^2 + 2x = -2\). First, calculate \((-1+i)^2\) and \(2(-1+i)\) separately.

Step 2: Calculate the Square

Calculate the square of \(-1+i\):\[(-1+i)^2 = (-1)^2 + 2(-1)(i) + i^2 = 1 - 2i - 1 = -2i\]

Step 3: Calculate the Linear Term

Next, find \(2(-1+i)\):\[2(-1+i) = -2 + 2i\]

Step 4: Add the Results

Add the results from the previous steps to find \(x^2 + 2x\):\[(-2i) + (-2 + 2i) = -2\]

Step 5: Verify with Original Equation

The sum \(-2\) is indeed equal to the right side of the original equation \(-2\). This confirms that \(-1+i\) satisfies the equation \(x^2 + 2x = -2\) and is therefore a solution.

Key Concepts

Complex Numbers

Complex numbers are a fundamental part of intermediate algebra, especially when working with equations that do not have real solutions. These numbers consist of a real part and an imaginary part. The imaginary unit is denoted as \(i\), which is defined by the property \(i^2 = -1\). For example, the complex number \(-1+i\) represents a real part of \(-1\) and an imaginary part of \(i\).
Complex numbers can be combined and manipulated much like regular numbers. Basic operations such as addition, subtraction, and multiplication follow specific rules.

• When adding or subtracting complex numbers, combine the real parts and the imaginary parts separately.
• For multiplication, use the distributive property and apply the rule \(i^2 = -1\) as necessary.

These operations extend the ability to solve types of equations that would otherwise have no solution in the realm of real numbers. Complex numbers open up the possibilities for solutions, particularly in quadratic equations.

Solving Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally taking the form \(ax^2 + bx + c = 0\). One common method for solving them is the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), but sometimes other methods are easier or more convenient.
Some quadratic equations may have solutions involving complex numbers, especially when the discriminant \(b^2 - 4ac\) is negative. In these cases, solutions are not real but instead involve the imaginary unit \(i\).
In the context of our exercise, the quadratic equation is expressed as \(x^2 + 2x = -2\). The task is to find out if a complex number like \(-1+i\) satisfies this equation. By substituting \(-1+i\) for \(x\), we can evaluate both sides to check if the equation balances. This approach helps confirm whether given numbers are solutions to the equation being solved.

Substitution Method

The substitution method is a straightforward technique used to check if a given number is a solution to an equation. This method involves replacing the variable in an equation with a specific number to see if the equation holds true.
In the example provided, the complex number \(-1+i\) is substituted into the quadratic equation \(x^2 + 2x = -2\). By evaluating \((-1+i)^2\) and \(2(-1+i)\), we find the resulting expressions:

• \((-1+i)^2 = -2i\)
• \(2(-1+i) = -2 + 2i\)

Adding these results together gives \(-2\), which matches the right side of the equation. This successful substitution confirms \(-1+i\) as a solution.
Substitution is particularly useful when verifying solutions of equations, whether those solutions are real or complex. It provides a clear, systematic approach to ensure that calculations are correct and that the proposed solution satisfies the original equation.