Problem 29 - Intermediate Algebra

Question

Use the method of completing the square to solve each quadratic equation. $$ x^{2}+5 x+1=0 $$

Step-by-Step Solution

Answer

Solutions are \( x = -\frac{5 + \sqrt{21}}{2} \) and \( x = -\frac{5 - \sqrt{21}}{2} \).

Step 1: Move the constant term to the right side

Start with the original equation:\[ x^2 + 5x + 1 = 0 \]Subtract 1 from both sides to isolate the quadratic and linear terms:\[ x^2 + 5x = -1 \]

Step 2: Find the term to complete the square

Take the coefficient of the linear term (5), divide it by 2, and square it:\[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]Add this square to both sides of the equation to maintain equality. The equation becomes:\[ x^2 + 5x + \frac{25}{4} = -1 + \frac{25}{4} \]

Step 3: Simplify and rewrite as a perfect square

Simplify the right side of the equation:\[ -1 + \frac{25}{4} = \frac{-4}{4} + \frac{25}{4} = \frac{21}{4} \]Now rewrite the left side as a perfect square:\[ \left(x + \frac{5}{2}\right)^2 = \frac{21}{4} \]

Step 4: Solve the equation

Take the square root of both sides:\[ x + \frac{5}{2} = \pm \sqrt{\frac{21}{4}} \]Remember that the square root of \(\frac{21}{4}\) is \( \frac{\sqrt{21}}{2} \), so:\[ x + \frac{5}{2} = \pm \frac{\sqrt{21}}{2} \]

Step 5: Isolate x to find the solutions

Subtract \( \frac{5}{2} \) from both sides:\[ x = -\frac{5}{2} \pm \frac{\sqrt{21}}{2} \]This results in two solutions:\[ x = -\frac{5 + \sqrt{21}}{2} \] and \[ x = -\frac{5 - \sqrt{21}}{2} \]

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations. By making a quadratic equation into a perfect square trinomial, this process allows us to find the solutions more easily. For the equation \[ x^2 + 5x + 1 = 0 \]we first need to isolate the quadratic and linear terms, resulting in: \[ x^2 + 5x = -1 \]Next, we determine the completion term; take half of the linear coefficient (5), divide it by 2, and square it. This gives us \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]We add this value to both sides to maintain balance, transforming the left side into a perfect square trinomial. We now rewrite the left side as \[ \left(x + \frac{5}{2}\right)^2 \]and adjust the right side to reflect these changes, allowing us to solve for the variable easily.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In particular, quadratic equations, which can be written in the standard form \( ax^2 + bx + c = 0 \), are a fundamental topic in algebra.Handling these equations often requires:

• Identifying coefficients \(a\), \(b\), and \(c\)
• Recognizing patterns and structure in equations
• Using algebraic operations like addition, subtraction, and factoring

These skills become crucial for strategies like completing the square, where knowledge of arithmetic sequences helps form perfect squares from quadratic terms.So when converting the quadratic equation, understanding each part in algebraic terms helps grasp why you add or manipulate components for a clearer solution. With algebraic principles, the quadratic transformation becomes a structured approach to isolating \(x\).

Step-by-Step Solutions

For solving quadratic equations, a structured, step-by-step approach can clarify the process. Let's break down the steps:1. **Isolate Terms**: First, separate relevant terms by moving the constant away. This paves the way for forming the square.2. **Calculate Completion**: Determine the completing number by dividing the linear term's coefficient by 2 and squaring it. Add this to both sides.3. **Form Perfect Square**: Convert the left side into a squared form, simplifying the equation structure.4. **Solve for x**: Take the square root of both sides, remembering to consider both positive and negative roots, the core aspect of squaring.5. **Find Solutions**: Finally, isolate \(x\) by performing arithmetic operations, deciphering two possible solutions due to the nature of quadratic equations.This methodology ensures clarity, allowing you to check your work at each stage. Approaching problems in this way demystifies the process and aids in understanding for future algebraic challenges.