In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. x^2 - 2x + 1 = 0

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1. Identify the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -2 \), and \( c = 1 \).
2. Use the formula for the discriminant \( \Delta = b^2 - 4ac \).
3. Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula: \( \Delta = (-2)^2 - 4 \times 1 \times 1 \).
4. Simplify the expression to find the value of the discriminant.
5. Determine the number and type of solutions based on the discriminant: if \( \Delta > 0 \), there are two distinct real solutions; if \( \Delta = 0 \), there is one real solution (a repeated root); if \( \Delta < 0 \), there are two complex solutions.

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 categorized 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. Recognizing these types is crucial for predicting the behavior of the quadratic function and understanding its graph, which can intersect the x-axis at different points depending on the nature of the solutions.