Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problema 79a

Capitolo 2, Problema 79a

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

Guida verificata passo dopo passo

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.

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.

Substitute the values of a, b, and c into the discriminant formula: Δ = (-2)^2 - 4(1)(1).

Simplify the expression for the discriminant. First, calculate (-2)^2, then calculate 4(1)(1), and finally subtract the second result from the first.

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.

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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.

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The Discriminant

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.

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Introduction to Quadratic Equations

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.

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Types of Slope

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