Compute the discriminant. Then determine the number and type of solutions for the given equation. x² - 2x + 1 = 0

Verified step by step guidance

1. Identify the coefficients of the quadratic equation in standard form, which is ax^2 + bx + c = 0. For the given equation x^2 - 2x + 1 = 0, the coefficients are: a = 1, b = -2, and c = 1.
2. Recall the formula for the discriminant, which is Δ = b^2 - 4ac. The discriminant helps determine the number and type of solutions for a quadratic equation.
3. Substitute the values of a, b, and c into the discriminant formula: Δ = (-2)^2 - 4(1)(1).
4. Simplify the expression for the discriminant. First, calculate (-2)^2, then calculate 4(1)(1), and finally subtract the second result from the first.
5. Interpret the value of the discriminant: If Δ > 0, there are two distinct real solutions. If Δ = 0, there is exactly one real solution (a repeated root). If Δ < 0, there are two complex solutions. Use this interpretation to determine the number and type of solutions for the given equation.

Key Concepts

Discriminant

The discriminant is a key component of the quadratic formula, given by the expression b² - 4ac for a quadratic equation in the form ax² + bx + c = 0. It helps determine the nature of the roots of the equation. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution (a repeated root); and if it is negative, there are two complex solutions.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, typically expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for analyzing their solutions.

Types of Solutions

The types of solutions for a quadratic equation are classified based on the value of the discriminant. Real solutions occur when the discriminant is non-negative, while complex solutions arise when the discriminant is negative. This classification is crucial for understanding the behavior of the graph of the quadratic function, which can intersect the x-axis at different points depending on the nature of the solutions.