Solve each equation by completing the square. 3 x^2=1-4 x |...

Name: Problem 55, Chapter 1: College Algebra
Uploaded: 2020-07-13T14:19:47-08:00
Duration: 3 min 44 s
Channel: Susan Cooper
Description: Solve each equation by completing the square. 3 x^2=1-4 x

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is an expression that can be written as the square of a binomial; for example, (x + d)² expands to x² + 2dx + d². Recognizing or forming a perfect square trinomial is the crux of the completing the square method, as it allows one to express a quadratic expression in a compact, squared form that is easier to solve or analyze.

Completing the Square

Completing the square is a method used to solve quadratic equations or to derive the vertex form of a parabola. This technique involves rewriting a quadratic expression as the square of a binomial plus or minus a constant. The process typically includes factoring out the coefficient of x² (if it is not 1), adding and subtracting the appropriate term to form a perfect square trinomial, and then solving the resulting equation.

Standard Form

Writing a quadratic equation in standard form (ax² + bx + c = 0) is a key step that makes it easier to analyze and solve the equation. Transforming the equation into this form involves rearranging terms so that all variables and constants are on one side of the equation, which is essential for methods like completing the square or the quadratic formula.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation typically written in the form ax² + bx + c = 0, where a, b, and c are constants and a ? 0. These equations represent parabolic graphs and have two solutions that can be real or complex, which are found by methods such as factoring, using the quadratic formula, or completing the square.