Solve each equation using the quadratic formula. See Examples 5 and 6. x^2 - 3x - 2 = 0

Verified step by step guidance

1. Identify the coefficients in the quadratic equation \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -3 \), and \( c = -2 \).
2. Write down 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: \( x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} \).
4. Simplify the expression under the square root (the discriminant): \( (-3)^2 - 4 \cdot 1 \cdot (-2) \).
5. Calculate the two possible values for \( x \) using the plus and minus in the quadratic formula: \( x = \frac{3 \pm \sqrt{\text{discriminant}}}{2} \).

Key Concepts

Quadratic Equation

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. The solutions to this equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula.

Quadratic Formula

The quadratic formula is a mathematical formula used to find the roots of a quadratic equation. It is expressed as x = (-b ± √(b² - 4ac)) / (2a). This formula provides a systematic way to calculate the values of x that satisfy the equation, regardless of whether the equation can be factored easily.

Discriminant

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots of the quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots.