Problem 64 - Elementary Algebra

Question

Solve by completing the square. $$(x+1)(x+7)-4(3 x+2)=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 2 + \sqrt{5}\) and \(x = 2 - \sqrt{5}\).

Step 1: Expand the Equation

First, expand the expression on the left side of the equation. \((x+1)(x+7)\) expands to \(x^2 + 7x + x + 7 = x^2 + 8x + 7\).Meanwhile, \(-4(3x+2)\) expands to \(-12x - 8\). Now, combine these terms: \(x^2 + 8x + 7 - 12x - 8 = 0\). Simplify by combining like terms: \(x^2 - 4x - 1 = 0\).

Step 2: Move Constant to the Other Side

To begin completing the square, start by isolating the constant on one side. Add 1 to both sides of the equation to achieve this: \(x^2 - 4x = 1\).

Step 3: Complete the Square

To complete the square, take half of the coefficient of the linear term \(-4\), which is \(-2\), and square it to get \(4\).Add \(4\) to both sides of the equation:\(x^2 - 4x + 4 = 1 + 4\). This transforms the equation into:\((x - 2)^2 = 5\).

Step 4: Solve for x

Now solve for \(x\) by taking the square root of both sides. This gives \(x - 2 = \pm \sqrt{5}\). Thus, solving for \(x\) leads to two possible solutions: \(x = 2 + \sqrt{5}\) or \(x = 2 - \sqrt{5}\).

Key Concepts

Quadratic Equations

Quadratic equations are a specific type of polynomial equation in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations are called quadratic because the highest power of the variable, usually \(x\), is squared.
Quadratic equations have several distinctive features and can be solved using different methods such as:

• Factoring
• Using the quadratic formula
• Completing the square

Completing the square transforms a standard quadratic equation into a perfect square trinomial, making it easier to solve. The term 'quadratic' originates from the Latin word 'quadratus,' meaning square, reflecting the squared variable present in the expression. When you solve a quadratic equation, you can find two possible solutions for \(x\), representing the points where the parabola intersects the x-axis in a graph.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In the equation \((x+1)(x+7) - 4(3x+2)\), we see two separate algebraic expressions combined through subtraction.
The steps in the solution require expanding these expressions by using distributive properties:
- Distributing \((x+1)\) with \((x+7)\), and - Distributing \(-4\) with \((3x+2)\)
This process allows you to rearrange terms systematically, eventually simplifying them into a form resembling the general structure of a quadratic equation. Understanding algebraic expressions well is essential, as this knowledge allows you to manipulate, expand, and simplify to uncover solutions effectively.

Mathematical Problem Solving

Mathematical problem solving involves breaking down and understanding a given problem, selecting suitable methods or strategies, executing those strategies, and ultimately finding the solution. In this exercise, solving by completing the square requires a series of methodical steps.
Each stage of solving the equation is meant to simplify or transform the problem into something more manageable. Here's how to engage in successful problem solving:

• Carefully read and interpret the problem statement.
• Break down the problem into smaller, more manageable parts.
• Select appropriate mathematical tools or strategies: like completing the square here.
• Perform operations step-by-step, double-checking for accuracy.
• Review the solution to ensure it meets the problem’s criteria.

The goal is not just to arrive at the answer, but to understand the process, which builds more robust problem-solving skills over time. By mastering methods such as completing the square, you enhance your ability to tackle complex algebraic problems.