Solve each equation in Exercises 83–108 by the method of your choice. 2x² - x = 1

Verified step by step guidance

1. Rewrite the equation in standard quadratic form by moving all terms to one side of the equation: $2x^2\; -\; x\; -\; 1\; =\; 0$.
2. Identify the coefficients of the quadratic equation in the form $ax^2\; +\; bx\; +\; c\; =\; 0$. Here, $a\; =\; 2$, $b\; =\; -1$, and $c\; =\; -1$.
3. Choose a method to solve the quadratic equation. For example, you can use the quadratic formula: $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$.
4. Substitute the values of $a$, $b$, and $c$ into the quadratic formula: $x\; =\; \backslash frac\{-(-1)\; \backslash pm\; \backslash sqrt\{(-1)^2\; -\; 4(2)(-1)\}\}\{2(2)\}$.
5. Simplify the expression under the square root (the discriminant) and then simplify the entire formula to find the two possible solutions for $x$.

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 properties of quadratic equations is essential for solving them effectively.

Factoring

Factoring involves rewriting an expression as a product of its factors. For quadratic equations, this often means expressing the equation in the form (px + q)(rx + s) = 0. This method is particularly useful when the quadratic can be easily factored, allowing for straightforward solutions by setting each factor equal to zero.

The Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method to find the roots of any quadratic equation. It is derived from completing the square and is applicable even when the equation cannot be factored easily. Understanding how to apply this formula is crucial for solving quadratic equations, especially when dealing with complex or irrational solutions.