Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x^2+4x−1=0

Verified step by step guidance

1. Identify the type of equation: This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 4 \), and \( c = -1 \).
2. Use the quadratic formula to solve for \( x \): \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
3. Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula: \( x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \).
4. Simplify the expression under the square root (the discriminant): \( 4^2 - 4 \cdot 1 \cdot (-1) \).
5. Continue simplifying the expression to find the values of \( x \).

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 structure of quadratic equations is essential for identifying their solutions.

Factoring

Factoring involves rewriting an expression as a product of its factors. For quadratic equations, this means expressing the equation in a form that can be easily solved, such as (x + p)(x + q) = 0. This method is particularly useful when the quadratic can be factored into integers, allowing for straightforward identification of the roots.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a systematic way to find the roots of any quadratic equation. It is derived from completing the square and is applicable even when factoring is not feasible. Understanding how to apply this formula is crucial for solving quadratic equations efficiently.