Problem 26 - College Algebra

Question

Solve each equation in Exercises \(15-26\) by the square root method. $$ (2 x+8)^{2}=27 $$

Step-by-Step Solution

Answer

The solutions for the equation \( (2x + 8)^2 = 27 \) are \( x = -0.132 \) and \( x = -4.868 \).

Step 1: Isolate the square

In the given equation, \( (2x + 8)^2 = 27 \), \( (2x + 8)^2 \) is already isolated. So, we can proceed to the next step.

Step 2: Apply the Square Root on both sides

Square root should be applied to both sides of the equation. This would give two sub-problems: \( 2x + 8 = \sqrt{27} \) and \( 2x + 8 = -\sqrt{27} \), because square root has two values, positive and negative.

Step 3: Solve for \( x \) in both equations

We now have two separate equations, and we solve each for \( x \):\[First Equation: 2x = \sqrt{27} - 8 \rightarrow x = \frac{\sqrt{27} - 8}{2}\]\[Second Equation: 2x = -\sqrt{27} - 8 \rightarrow x = \frac{-\sqrt{27} - 8}{2}\]

Step 4: Simplify the answers

Finally, we simplify each equation to the lowest terms possible. With the help of a calculator, we obtain two solutions: \( x = -0.132 \) and \( x = -4.868 \).

Key Concepts

Solving Equations

Solving equations is a foundational skill in algebra, helping you to find the value of unknown variables that make the equation true. The heart of solving an equation is balancing both sides, akin to keeping a scale level. Here, our equation \( (2x+8)^2 = 27 \) follows this principle.

• First, ensure the part to be solved is isolated on one side.
• Next, perform the same mathematical operation on both sides to maintain the balance.

Remember, solving quadratic equations, like in our example, often reveals two solutions due to the nature of squaring. Each step systematically uncovers these possibilities.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written as \( ax^2 + bx + c = 0 \). They arise frequently in real-world scenarios and have distinctive characteristics:

• They produce parabolic graphs, either opening upwards or downwards.
• The solutions, or roots, of quadratics can be found via factoring, completing the square, or using the quadratic formula.
• In this exercise, we apply the square root method, suitable when the equation can be neatly expressed as \( (px+q)^2 = r \).

Understanding the format and finding the roots efficiently can aid in solving real-world problems and mathematical puzzles alike. Solving \( (2x+8)^2 = 27 \) showcases how equating to a perfect square simplifies the task.

Algebraic Techniques

Algebraic techniques vary and are vital tools when solving equations. They include operations and strategies applied to both linear and quadratic equations, among others. Our example demonstrates several key algebraic techniques:

• Isolation techniques to separate the \( x \) term, ensuring clarity in handling subsequent steps.
• Applying the square root to both sides safely transitions quadratic expressions, revealing two potential solutions.
• Simplifying expressions leverages tools like distributive properties and root simplifications, aiding in exact and approximate solutions.

Mastery over these techniques allows one to approach complex problems more strategically, ultimately leading to easier and faster problem-solving.