Use the method of completing the square to solve each...

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It has the form (x + d)², which expands to x² + 2dx + d². Recognizing and creating perfect square trinomials is a key step in the method of completing the square, as it simplifies the process of solving quadratic equations and analyzing quadratic functions.

Square Root Principle

The square root principle involves taking the square root of both sides of an equation. Once a quadratic equation has been rewritten in the form (x + d)² = e, the square root principle allows you to solve for x by taking the positive and negative square roots of e, thereby yielding the solutions to the equation. This principle is foundational when working with equations that have been simplified through completing the square.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in one variable, typically expressed in the form ax² + bx + c = 0. These equations can be solved using various techniques, including factoring, applying the quadratic formula, graphing, or completing the square. Understanding the structure of a quadratic equation is essential for identifying the appropriate method for finding its roots.

Completing the Square

Completing the square is an algebraic technique used to solve quadratic equations by transforming the quadratic expression into a perfect square trinomial. This is achieved by rearranging terms and adding a specific constant to both sides of the equation, which allows the quadratic to be written as the square of a binomial. This method is also useful in understanding the properties of quadratic functions, such as their vertex form and axis of symmetry.

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