Factor completely. 4 x^2+x+9

Name: Problem 65, Chapter 13: Prealgebra & Introductory Algebra
Uploaded: 2022-04-03T15:16:39-08:00
Duration: 14 s
Channel: Erika Bustos
Description: Factor completely. 4 x^2+x+9

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Key Concepts

Quadratic Polynomials

Quadratic polynomials are expressions of the form ax² + bx + c where a, b, and c are constants and a ? 0. Factoring these involves finding a product of two binomials that results in the original quadratic, which typically requires identifying two numbers whose product equals a×c and whose sum equals b.

Discriminant Analysis

The discriminant of a quadratic, calculated as b² - 4ac, is essential in determining the nature of its roots. A positive perfect square discriminant indicates that the quadratic can be factored over the integers, while a negative or non-perfect square discriminant suggests that the roots are irrational or complex, making factorization into linear factors with integer coefficients impossible.

Factorization over Different Number Sets

Sometimes a quadratic polynomial does not factor completely over the integers even though it may factor over the real or complex numbers. Recognizing whether factorization is possible within a specific number set—integers, reals, or complexes—is a key skill when working with quadratic expressions.

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