Solve each equation using the quadratic formula. x² - 3x - 2 = 0

Verified step by step guidance

1. Identify the coefficients in the quadratic equation \(x^2 - 3x - 2 = 0\). Here, \(a = 1\), \(b = -3\), and \(c = -2\).
2. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
3. Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-2)}}{2(1)}\).
4. Simplify inside the square root (the discriminant): calculate \((-3)^2 - 4(1)(-2)\).
5. Write the expression for \(x\) with the simplified discriminant under the square root, and express the two possible solutions using the \(\pm\) symbol.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the coefficients.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method to find the roots of any quadratic equation. It uses the coefficients a, b, and c to calculate the solutions directly, especially useful when factoring is difficult or impossible.

Discriminant

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