Question

Solve the equation using any method. Explain your reasoning. \(-2(x+2)^2=5\)

Short answer

The roots of the equation are \(x = -2 + \sqrt{\frac{5}{2}}*i\) and \(x = -2 - \sqrt{\frac{5}{2}}*i\)

Step-by-step solution

Isolate the Quadratic Term

First, it is necessary to isolate the square term from the equation. Let's do this by dividing the whole equation by -2. It would then look like this: \((x+2)^2 = -\frac{5}{2}\)

Removing the Square

As the next step, to get rid of the square, calculate the square root on both sides. It will create two cases (positive and negative) as the double roots could give rise to two solutions. This will get us that: \(x+2 = \sqrt{-\frac{5}{2}}\) and \(x+2 = -\sqrt{-\frac{5}{2}}\)

Treat the square root of a negative number

Take note that the square root of a negative number is not a real number but an imaginary number. It is represented as \(\sqrt{-1} = i\). From the previous step we get that: \(x+2 = \sqrt{\frac{5}{2}}*i\) and \(x+2 = -\sqrt{\frac{5}{2}}*i\)

Solve for x

Having the equation as it is, solve for x for both roots by subtracting 2: \(x = -2 + \sqrt{\frac{5}{2}}*i\) and \(x = -2 - \sqrt{\frac{5}{2}}*i\)

Key concepts

Imaginary Numbers

Imaginary numbers might sound complex, but they are quite straightforward once you grasp the basic concept. They are called "imaginary" because they involve the square root of a negative number—a concept that doesn't exist in the set of real numbers. When you take the square root of a negative number, you'll use the imaginary unit, represented by the symbol \(i\), where \(i = \sqrt{-1}\). This idea extends the number system you are familiar with.

• The product of \(i\) with itself is \(i^2 = -1\).
• Any real number multiplied by \(i\) becomes an imaginary number like \(3i, 4.5i,\) or \(-0.7i\).

To work with these numbers, especially when solving equations, it's essential to integrate both real and imaginary parts, called complex numbers, typically expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. These complex numbers are vital in various fields like engineering and physics.

Quadratic Equations

Quadratic equations play a critical role in mathematics and usually take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The equation from the exercise, once simplified, is slightly different and doesn't look like the standard form. However, it is a quadratic equation because it features an \(x^2\) term.

• Quadratic equations can be solved using various methods, including factoring, using the quadratic formula, completing the square, or graphing.
• In the original exercise, isolating the quadratic term makes it easier to apply the square root method.

Understanding how to handle quadratic equations is crucial because they appear frequently in algebraic problems. The solutions to such equations might be real or involve imaginary numbers, as seen when dealing with negative square roots.

Solving Equations

When solving equations, the goal is to find the values satisfying the equality. The exercise uses a structured approach to break down and solve the given quadratic equation:

• Isolation: Start by isolating the quadratic term. This simplification allows you to deal directly with the expression on one side of the equation.
• Square Roots: Taking the square root helps in reducing the quadratic equation to a linear form by introducing potential solutions, both positive and negative.
• Handling Imaginary Results: Since solutions sometimes involve negatives under a square root, imaginary unit \(i\) becomes essential. This approach moves you from real to complex solutions.
• Final Solution: Examine both solutions separately to solve for \(x\).

Each step brings you closer to solving the equation, demonstrating the importance of understanding both real and complex numbers. This structured breakdown is valuable, ensuring you carefully handle each part of the equation to get to the correct solutions.