Question

Solve \(2(x+1)^{2}+8=0\) in the complex number system.

Short answer

The solutions are \(x = -1 + 2i\) and \(x = -1 - 2i\).

Step-by-step solution

- Simplify the equation

First, simplify the given equation: \[2(x+1)^{2}+8=0\]

- Isolate the square term

Subtract 8 from both sides to isolate the square term: \[2(x+1)^{2} = -8\]

- Divide by the coefficient

Divide both sides by 2 to simplify further: \[(x+1)^{2} = -4\]

- Take the square root

Take the square root of both sides, remembering to include the imaginary unit \(i\): \[x+1 = \pm 2i\]

- Solve for x

Subtract 1 from both sides to solve for \(x\): \[x = -1 \pm 2i\]

Key concepts

Complex Numbers

Complex numbers extend the idea of one-dimensional numbers (real numbers) to two dimensions by incorporating an imaginary unit. This allows for the representation of numbers that include a combination of a real part and an imaginary part, written as \(a + bi\). Here, \(a\) and \(b\) are real numbers, while \(i\) is the imaginary unit. Complex numbers are essential when solving equations that do not have real solutions, such as quadratic equations that yield negative square roots.

Square Roots

The square root of a number is another number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\). However, when dealing with negative numbers under the square root, we introduce imaginary units. For example, the square root of -4 is written as \(2i\), because \(i^2 = -1\). Remember to always consider both the positive and negative square roots: \( \sqrt{-4} = \pm 2i \).

Isolating Variables

Isolating variables is a crucial step in solving equations. It involves manipulating the equation to have the variable isolated on one side of the equation. For example, when you have the equation \(2(x+1)^{2} + 8 = 0\), subtracting 8 from both sides results in \(2(x+1)^{2} = -8\). From there, dividing by 2 isolates \((x+1)^{2}\) altogether. This ensures that solving equations becomes more straightforward and manageable.

Imaginary Unit

The imaginary unit, denoted as \(i\), is defined as \(i = \sqrt{-1}\). Taking the square of the imaginary unit gives \(i^2 = -1\). It's vital for handling operations involving square roots of negative numbers. Anytime you encounter a negative number under the square root, you use \(i\) to transform it into a manageable form. For instance, \( \sqrt{-4} = 2i\). This concept allows us to extend real numbers to complex numbers and find solutions to equations that appear unsolvable in the real number system.

Quadratic Equations

Quadratic equations are polynomials of the second degree, generally represented as \(ax^2 + bx + c = 0\). These can have two, one, or no real solutions depending on the discriminant \(b^2 - 4ac\). When the discriminant is negative, the solutions involve complex numbers. For example, solving \(2(x + 1)^{2} + 8 = 0\) involves steps like simplifying, isolating the variable term, and taking the square root of both sides. The solutions \(x = -1 \pm 2i\) are complex because the equation's discriminant results in a negative value under the square root.