91Ó°ÊÓ

Solving equations is a fundamental concept in mathematics, where the goal is to find the value or values of the variable that make the equation true. In this exercise, the equation we need to solve is \((r+10)^{2} = 4\). Let's break down the process:

- We have an equation where the expression \((r+10)\) is squared, and it's equal to 4. The task is to find 'r.' - One way to solve equations like this is by using the square root property. This technique allows us to "undo" the square by taking the square root on both sides.

The key is to work step-by-step, carefully isolating terms and solving for the variable. It's worth noting that solving equations involves critical thinking and logical reasoning to systematically approach the problem while verifying each step.

Square roots are essential when it comes to solving equations that involve squared terms. A square root of a number is a value that, when multiplied by itself, gives the original number. Here's how square roots are used in solving the exercise:

- The equation \((r+10)^{2} = 4\) involves applying the square root property to simplify the process. This principle states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\).

When applying the square root to both sides:

• We get \((r+10) = \pm \sqrt{4}\).
• Recognize that \(\sqrt{4}\) simplifies to 2.

Now, having \((r+10) = \pm 2\) means there are two possible routes to follow, leading us to find two solutions.

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). However, not all quadratic equations appear in this standard form. The exercise we've solved is a special case of a quadratic equation, expressed as \((r+10)^{2} = 4\). Understanding these concepts helps:

- A key feature of quadratic equations is their potential to have two solutions, which is exactly what we found in this problem.
- By using the square root property, we derived two linear equations, \((r+10) = 2\) and \((r+10) = -2\), leading to solutions \(r = -8\) and \(r = -12\), respectively.This exercise wonderfully illustrates the principle that quadratic equations can often be solved through simplification and algebraic manipulation, sometimes requiring us to recognize a perfect square structure to apply the square root property effectively.