Solve each equation. x²- √5x -1 = 0

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + 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.

The Quadratic Formula

The quadratic formula, given by x = (-b ± √(b² - 4ac)) / (2a), provides a method to find the roots of any quadratic equation. The term under the square root, known as the discriminant (b² - 4ac), determines the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; and if negative, two complex roots. This formula is crucial for solving the given equation.

Radicals and Square Roots

Radicals involve roots of numbers, and the square root is a specific case where we find a number that, when multiplied by itself, gives the original number. In the context of the equation x² - √5x - 1 = 0, understanding how to manipulate and simplify expressions involving square roots is vital for isolating x and finding the solutions. Mastery of radicals is essential for solving equations that include them.