%5E2-roots.jpeg "(m-1)^2 Roots")
Understanding (m-1)^2 Roots
The expression (m-1)^2 represents the square of the difference between a variable 'm' and 1. Understanding its roots involves finding the values of 'm' that make the expression equal to zero. Let's break down this concept:
Roots and Solutions
- Roots are the values that make an equation equal to zero. In this case, we're looking for the roots of the equation: (m-1)^2 = 0.
- Solutions are another way of referring to the roots of an equation. They are the values that satisfy the equation.
Finding the Roots
- Expand the equation: (m-1)^2 = (m-1)(m-1) = m^2 - 2m + 1
- Set the equation to zero: m^2 - 2m + 1 = 0
- Factor the equation: (m-1)(m-1) = 0
- Solve for 'm': (m-1) = 0, which gives us m = 1.
Understanding the Solution
This means that the equation (m-1)^2 = 0 has only one root or one solution, which is m = 1. This is because the equation represents a perfect square, and squaring any value results in a non-negative number. The only way for the square to be zero is if the base itself is zero, in this case, (m-1) = 0.
Visualizing the Solution
You can visualize this by plotting the graph of the function y = (m-1)^2. The graph will be a parabola with its vertex at (1,0). The vertex represents the point where the function intersects the x-axis, and this point corresponds to the root of the equation.
Key Takeaways
- The equation (m-1)^2 = 0 has only one root, which is m = 1.
- This root represents the value that makes the equation true, resulting in a value of zero for the expression.
- Understanding roots is fundamental in solving equations and analyzing functions.
By understanding the concept of roots and their relationship to equations, you gain valuable insights into the behavior of mathematical expressions and can effectively solve problems in various areas of mathematics.