Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by completing the square. $$ x^{2}+2 x-8=0 $$

Step-by-Step Solution

Answer

The short answer to the given problem: Rearrange the equation \(x^2 + 2x = 8\). Complete the square by adding and subtracting 1: \((x^2 + 2x + 1) - 1 = 8\). Rewrite and simplify as \((x + 1)^2 = 9\). Take the square root of both sides to get \(x + 1 = \pm3\). Solve for x, which gives two solutions: \(x = 2, -4\).

Step 1: Rearrange the equation

Before completing the square, let's rearrange the equation and move the constant to the other side: $$ x^2 + 2x = 8 $$

Step 2: Add and subtract the square of half of the coefficient of x

The coefficient of x is 2. We take half of it, which is 1, and square it to get 1. Then we add and subtract 1 on the left side of the equation to complete the square: $$ (x^2 + 2x + 1) - 1 = 8 $$

Step 3: Rewrite as a square and simplify

Now we can rewrite the expression in parentheses as a square and simplify the equation: $$ (x + 1)^2 - 1 = 8 $$

Step 4: Move the constant to the other side

Now let's move -1 to the right side of the equation and continue simplifying: $$ (x + 1)^2 = 9 $$

Step 5: Take square root of both sides

Now, to solve for x, we can take the square root of both sides of the equation: $$ \sqrt{(x + 1)^2}=\pm\sqrt{9} $$

Step 6: Solve for x

Simplify and solve for x: $$ x + 1 = \pm3 $$ There are two cases: 1) When x + 1 = 3, we have x = 2. 2) When x + 1 = -3, we have x = -4. So, the solutions to the equation are: $$ x = 2, -4 $$

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest degree of the variable is two. These equations take the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a eq 0 \). Solving quadratic equations involves finding the values of \( x \) that make the equation true. These values are called the roots or solutions of the equation.

Quadratic equations can be solved through various methods including factoring, using the quadratic formula, graphing, and a process known as "completing the square."

This method, as demonstrated in the exercise, involves rewriting the equation to form a perfect square trinomial on one side. This approach is particularly useful when the quadratic does not factor easily. Completing the square transforms the quadratic equation into one that is easier to solve by roots extraction.

Solving Equations

Solving equations means finding the value or set of values that satisfy the given equation. When solving quadratic equations, the goal is to find the roots or solutions.

The process of completing the square involves several steps:

• Rearrange: First, manipulate the equation to place the quadratic and linear terms on one side and the constant term on the other.
• Complete the Square: Add and subtract the square of half the coefficient of the linear term. This transforms the expression into a perfect square trinomial.
• Square Root Extraction: Take the square root of both sides to solve for the variable.

Each step transforms the equation into a simpler form, guiding us steadily towards the solution. Completing the square is a crucial skill, sharpening one's algebraic manipulation abilities and understanding of quadratic relationships.

Mathematical Methods

Mathematical methods for solving equations, like completing the square, showcase the variety and complexity of algebraic problem-solving techniques. Each method has its own set of procedures and advantages.

The strengths of completing the square lie in its ability to simplify quadratic equations into forms for which direct square root analysis can be applied. It introduces students to deeper algebraic concepts, such as:

• Perfect square trinomials: Recognizing and forming expressions like \((x + 1)^2\) from \(x^2 + 2x + 1\).
• Equation Reordering: Strategically moving terms across the equation to facilitate solving.
• Handling Radicals: Understanding the implications of square roots and their solutions.

Understanding and mastering these methods enhances mathematical intuition and problem-solving efficiency, helping students approach complex problems with confidence.