Solve each equation in Exercises 65–74 using the quadratic formula. 3x^2 - 3x - 4 = 0

Verified step by step guidance

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

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), where a, b, and c are the coefficients from the quadratic equation. This formula provides a systematic way to calculate the solutions, even when the equation cannot 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.