Problem 41 - Elementary Algebra

Question

\(m^{2}-2 m+1=0\)

Step-by-Step Solution

Answer

The solution is \(m = 1\).

Step 1: Identify the equation

The given equation is a quadratic equation: \(m^{2}-2m+1=0\).

Step 2: Rewrite in standard form (if necessary)

The equation is already in standard form \(am^2 + bm + c = 0\) with \(a = 1\), \(b = -2\), and \(c = 1\).

Step 3: Check if it can be factored

The quadratic equation can be factored as \((m-1)^2 = 0\).

Step 4: Solve the factored equation

The factored equation \((m-1)^2 = 0\) gives the solution \m - 1 = 0\. Solving this gives \m = 1\.

Step 5: Verify the solution

Substitute \(m = 1\) back into the original equation to verify: \(1^2 - 2 \times 1 + 1 = 0\) simplifies to \(0 = 0\), confirming the solution is correct.

Key Concepts

Factoring Quadratics

Factoring quadratics is a key technique that helps us solve quadratic equations efficiently. When we factor a quadratic equation, we rewrite it as a product of two binomials.
For example, consider the quadratic equation in our exercise: \(m^2 - 2m + 1 = 0\).
This expression can be factored as: \((m-1)(m-1) = 0\) or in simplified form, \((m-1)^2 = 0\).
Factoring quadratics is useful because it transforms a complex equation into a simpler one that is easier to solve.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). In this format, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.
For our given equation \(m^2 - 2m + 1 = 0\), we can identify:

• \(a = 1\)
• \(b = -2\)
• \(c = 1\)

This makes it easier to perform operations such as factoring or applying the quadratic formula.
Always ensure your quadratic equation is in the standard form before attempting to solve it.

Quadratic Equation Solutions

To find the solutions to a quadratic equation, we can use various methods including factoring, completing the square, or using the quadratic formula. For our equation \((m-1)^2 = 0\), once factored, we solve it by setting each factor equal to zero:

• \(m-1 = 0\)

Solving this, we get \(m = 1\). Quadratic equations can have multiple solutions depending on the degree and the discriminant.
Always double-check your solutions by substituting them back into the original equation to verify their correctness.