We need to solve the quadratic equation x squared minus 2x plus 1 equals 0. This equation is already in standard form ax squared plus bx plus c equals 0, where a equals 1, b equals negative 2, and c equals 1. We will solve this equation using three different algebraic methods to find the value or values of x that make the equation true.
Let's solve this equation using the factoring method. First, we recognize that x squared minus 2x plus 1 is a perfect square trinomial. We can factor this as x minus 1 quantity squared. So our equation becomes x minus 1 quantity squared equals 0. Taking the square root of both sides, we get x minus 1 equals 0, which gives us x equals 1. This is a repeated root with multiplicity 2, meaning the factor x minus 1 appears twice.
Now let's verify our solution using the quadratic formula. For any quadratic equation ax squared plus bx plus c equals 0, the solution is x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. With our coefficients a equals 1, b equals negative 2, and c equals 1, we first calculate the discriminant: b squared minus 4ac equals negative 2 squared minus 4 times 1 times 1, which equals 4 minus 4, equals 0. When the discriminant equals zero, we have exactly one solution. Substituting into the formula: x equals 2 plus or minus square root of 0, all divided by 2, which simplifies to x equals 1. This confirms our factoring result.
Let's solve this equation using the completing the square method. We start with x squared minus 2x plus 1 equals 0. First, we rearrange to get x squared minus 2x equals negative 1. To complete the square, we take half of the coefficient of x, which is negative 2 divided by 2, equals negative 1. Then we square this result: negative 1 squared equals 1. However, we notice that we already have plus 1 on the left side of our original equation. This means x squared minus 2x plus 1 is already a perfect square: x minus 1 quantity squared. So our equation becomes x minus 1 quantity squared equals 0, which gives us x equals 1. This method reveals why the original expression was a perfect square trinomial.
Let's verify our solution graphically by plotting the parabola y equals x squared minus 2x plus 1. This parabola opens upward since the coefficient of x squared is positive. The key observation is that the parabola touches the x-axis at exactly one point: the point 1, 0. This point is both the vertex of the parabola and the x-intercept. Since the parabola just touches the x-axis without crossing it, this confirms that x equals 1 is a repeated root. The x-intercept represents the solution to our original equation x squared minus 2x plus 1 equals 0. This graphical method provides visual confirmation of our algebraic solutions.