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

Verified step by step guidance

1. Start by moving all terms to one side of the equation to set it to zero: \(2x^2 - x - 1 = 0\).
2. Identify the quadratic equation in the standard form \(ax^2 + bx + c = 0\), where \(a = 2\), \(b = -1\), and \(c = -1\).
3. Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for \(x\).
4. Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2}\).
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. These equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the standard form and the properties 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.

The Quadratic Formula

The quadratic formula, given by x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of any quadratic equation. This formula is derived from the process of completing the square and is particularly useful when the equation cannot be easily factored. Knowing how to apply the quadratic formula is vital for solving quadratic equations that do not have rational roots.