Problem 77 - Elementary Algebra

Question

Solve by completing the square. \(m^{2}+4 m=-44\)

Step-by-Step Solution

Answer

The solutions are \( -2 \pm 2i \sqrt{10} \).

Step 1 - Move the constant to the right side

To start, rearrange the equation by moving the constant term to the right side. The equation becomes: \(m^2 + 4m = -44\)

Step 2 - Identify the coefficient of the linear term

Here, the coefficient of the linear term (\(m\)) is 4. You need this to complete the square.

Step 3 - Divide the coefficient of the linear term by 2 and square it

Divide the coefficient of \(m\) by 2, then square the result. \( \left( \frac{4}{2} \right)^2 = 4 \)

Step 4 - Add and subtract this square inside the equation

Add and subtract 4 on the left side of the equation to complete the square: \( m^2 + 4m + 4 - 4 = -44 \) This simplifies to: \( (m + 2)^2 - 4 = -44 \)

Step 5 - Isolate the perfect square

Move the constant term on the left to the right side to isolate the perfect square: \( (m + 2)^2 = -40 \)

Step 6 - Solve for \(m\)

Take the square root of both sides and solve for \(m\): \( m + 2 = \pm \sqrt{-40} \) Since \( \sqrt{-40} = \sqrt{-1 \cdot 4 \cdot 10} = 2i \sqrt{10} \), \( m = -2 \pm 2i \sqrt{10} \)

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, meaning they contain at least one term that is squared. The general form of a quadratic equation is: ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
The solutions to a quadratic equation are its roots, which can be real or complex numbers. Quadratic equations can be solved using:

• Factoring
• Completing the square
• The quadratic formula

In this exercise, we solved a quadratic equation by completing the square. This method turns a quadratic equation into a perfect square trinomial, simplifying the process.

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to the complex plane by adding a second dimension. A complex number is written in the form a + bi, where:

• a is the real part
• bi is the imaginary part

The imaginary unit, i, is defined by the property that i^2 = -1. This allows the square root of negative numbers to be defined.

In our exercise, after isolating the perfect square, we ended up with (m + 2)^2 = -40. Taking the square root of both sides gives us 2i √10, which includes the imaginary unit i. Therefore, the solutions to the quadratic equation also include complex numbers.

Algebraic Methods

Algebraic methods involve operations and manipulation of equations to find unknown values. Common algebraic methods include:

• Addition and subtraction
• Multiplication and division
• Factoring
• Expanding expressions
• Completing the square
• Using formulas, like the quadratic formula

Rewriting the equation by moving the constant term to the other side

Finding the value to complete the square

Adding and subtracting that value

Factoring and simplifying the equation

Solving for the variable

Understanding and mastering these steps is crucial for solving quadratic equations efficiently.