Solve each equation in Exercises 83–108 by the method of your choice. x^2 - 2x = 1

Verified step by step guidance

1. Start by rewriting the equation in standard quadratic form: $x^2-2x-1=0$.
2. Identify the coefficients for the quadratic formula: $a=1$, $b=-2$, and $c=-1$.
3. Use the quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.
4. Substitute the values of $a$, $b$, and $c$ into the formula: $x=\frac{-(-2)\pm \sqrt{(-2{)}^2-4\cdot 1\cdot (-1)}}{2\cdot 1}$.
5. Simplify the expression under the square root and solve for $x$ using the quadratic formula.

Key Concepts

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 structure of quadratic equations is essential for solving them effectively.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. For quadratic equations, this often involves rewriting the equation in a form that can be expressed as a product of two binomials. Mastery of factoring techniques is crucial for quickly solving quadratic equations.

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This technique involves manipulating the equation to isolate the variable on one side, allowing for easier solution finding. It is particularly useful when the quadratic does not factor neatly, providing an alternative approach to finding the roots.