Solve each equation using completing the square. See Examples 3 and 4. x^2 - 2x - 2 = 0

Verified step by step guidance

1. Start with the equation: $x^2 - 2x - 2 = 0$.
2. Move the constant term to the other side of the equation: $x^2 - 2x = 2$.
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 this square to both sides of the equation: $x^2 - 2x + 1 = 2 + 1$.
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 the equation into a perfect square trinomial. This involves rearranging the equation and adding a specific value to both sides to create a square of a binomial. This technique simplifies the process of finding the roots of the equation and is particularly useful when the quadratic formula is not preferred.

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 ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the standard form and properties of quadratic equations is essential for effectively solving them.

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 allows for easier manipulation of the equation to isolate the variable and find solutions.