Final answer:
The most efficient technique to solve the equation (2x+7)² = 36 is by using the square root method, which simplifies the quadratic to a linear equation and results in two possible solutions for x.
Step-by-step explanation:
The most efficient technique for solving (2x+7)² = 36 is by using the square root method. Instead of expanding the square or using the quadratic formula, we can take the square root of both sides of the equation. This simplifies the problem from a quadratic to a linear equation, which is much easier to deal with.
To solve the equation, follow these steps:
- Take the square root of both sides of the equation, giving us √(2x+7)² = √36.
- Simplify to obtain 2x + 7 = ±√36, which further simplifies to 2x + 7 = ±6.
- Now, solve the two linear equations separately: 2x + 7 = 6 and 2x + 7 = -6.
- Subtract 7 from both sides to isolate the term with x, resulting in 2x = -1 and 2x = -13 respectively.
- Finally, divide by 2 to solve for x, giving x = -0.5 and x = -6.5.