Solve each equation in Exercises 83–108 by the method of your choice. (3x - 4)² = 16

Verified step by step guidance

1. Start by recognizing that the equation \((3x - 4)^2 = 16\) is a quadratic equation in the form of a perfect square. To solve, take the square root of both sides of the equation. Remember to include both the positive and negative roots when taking the square root.
2. After taking the square root, the equation becomes \(3x - 4 = \pm 4\). This means you now have two separate linear equations to solve: \(3x - 4 = 4\) and \(3x - 4 = -4\).
3. Solve the first equation \(3x - 4 = 4\) by isolating \(x\). Add 4 to both sides to get \(3x = 8\), then divide both sides by 3 to find \(x\).
4. Solve the second equation \(3x - 4 = -4\) by isolating \(x\). Add 4 to both sides to get \(3x = 0\), then divide both sides by 3 to find \(x\).
5. Combine the solutions from both equations to express the final solution set. These are the values of \(x\) that satisfy the original equation.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the properties of quadratic equations is essential for solving them effectively.

Square Roots

Square roots are the values that, when multiplied by themselves, yield the original number. In the context of the equation (3x - 4)^2 = 16, taking the square root of both sides is a crucial step to isolate the variable. It is important to consider both the positive and negative roots when solving such equations.

Isolating Variables

Isolating variables involves rearranging an equation to solve for a specific variable. This process often includes using inverse operations, such as addition, subtraction, multiplication, or division, to get the variable alone on one side of the equation. Mastery of this concept is vital for effectively solving equations and understanding their solutions.