Solve each equation in Exercises 47–64 by completing the square. x^2 - 2x = 2

Verified step by step guidance

1. Start with the equation: \(x^2 - 2x = 2\).
2. Move the constant term to the other side to set up for completing the square: \(x^2 - 2x = 2\).
3. To complete the square, take half of the coefficient of \(x\), which is \(-2\), divide it by 2 to get \(-1\), and then square it to get \(1\).
4. Add \(1\) to both sides of the equation to maintain equality: \(x^2 - 2x + 1 = 2 + 1\).
5. Rewrite the left side as a perfect square trinomial: \((x - 1)^2 = 3\).

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial. This involves manipulating the equation so that one side becomes a squared binomial, allowing for easier solutions. For example, in the equation x^2 - 2x = 2, we would add and subtract the square of half the coefficient of x to both sides to facilitate this transformation.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not zero. The solutions to these equations can be found using various methods, including factoring, using the quadratic formula, or completing the square. Understanding the standard form of a quadratic equation is essential for applying these methods effectively.

Perfect Square Trinomial

A perfect square trinomial is an expression that can be factored into the square of a binomial, typically in the form (x + p)^2 = x^2 + 2px + p^2. Recognizing and creating perfect square trinomials is crucial when completing the square, as it simplifies the process of solving quadratic equations. For instance, transforming x^2 - 2x into (x - 1)^2 involves identifying the necessary constant to complete the square.