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A quadratic equation is a type of polynomial equation characterized by having a degree of 2. This simply means that the highest exponent of the variable, which is often represented by \(x\), is 2. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) cannot be zero. When \(a\) is zero, the equation is no longer quadratic, but linear.

Solving a quadratic equation involves finding the values of \(x\) that make the equation true. These values are known as the roots of the equation. Quadratic equations are fundamental in algebra and can be solved using various methods, including factoring, completing the square, and using the quadratic formula.

• Factoring: Breaking down the quadratic into products of simpler expressions.
• Completing the Square: Transforming the equation into a perfect square trinomial.
• Quadratic Formula: An efficient method when the equation cannot easily be factored.

The discriminant is a crucial part of understanding quadratic equations, particularly in predicting the nature of the roots without solving the equation. It is the part of the quadratic formula expressed by \(b^2 - 4ac\). This value helps us determine how many and what type of solutions or roots the quadratic equation has.

Here's a quick guide to interpreting the discriminant:

• If the discriminant is positive \((b^2 - 4ac > 0)\), the quadratic equation has two distinct real roots. This means the parabola will intersect the x-axis at two different points.
• If the discriminant is zero \((b^2 - 4ac = 0)\), the equation has exactly one real root or a repeated root. Here, the parabola touches the x-axis at exactly one point.
• If the discriminant is negative \((b^2 - 4ac < 0)\), there are no real roots; instead, there are two complex roots. The parabola does not touch or cross the x-axis at all.

In the provided exercise, the discriminant is zero, so the equation has a single repeated root.

The roots of an equation are the values of \(x\) which satisfy the equation, effectively making it equal to zero. For quadratic equations, these roots can be real or complex, depending on the discriminant. In solving quadratic equations, finding the roots is the primary goal.

For the exercise \(x^2 - 2x + 1 = 0\), the discriminant is 0, indicating that there is one real, repeated root. We calculated this using the quadratic formula, leading to the root \(x = 1\).

• When solving, the roots are determined by evaluating the expressions given by the quadratic formula.
• If the roots are both the same, as in this problem, the solution might represent a point where the graph of the equation, a parabola, is tangent to the x-axis.
• Roots provide critical points of intersection for graphing, allowing the equation to be visualized easily.