Solve each equation. x²- √(5x) -1 = 0

Verified step by step guidance

1. Identify the given quadratic equation: \(x^{2} - \sqrt{5}x - 1 = 0\).
2. Recall the quadratic formula for solving equations of the form \(ax^{2} + bx + c = 0\): \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).
3. Determine the coefficients from the equation: \(a = 1\), \(b = -\sqrt{5}\), and \(c = -1\).
4. Substitute the coefficients into the quadratic formula: \(x = \frac{-(-\sqrt{5}) \pm \sqrt{(-\sqrt{5})^{2} - 4(1)(-1)}}{2(1)}\).
5. Simplify inside the square root and the numerator step-by-step to express the solutions for \(x\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0. Solving such equations involves finding values of x that satisfy the equation, often using methods like factoring, completing the square, or the quadratic formula.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a direct way to find the roots of any quadratic equation ax² + bx + c = 0. It uses the coefficients a, b, and c to calculate the solutions, including complex roots when the discriminant is negative.

Discriminant and Its Role

The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots. Understanding this helps predict the type of solutions before solving.