Solve each equation in Exercises 68–70 using the quadratic formula. 2x^2 = 3-4x

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. This equation represents a parabola when graphed. The solutions to the equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula.

The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation. It is expressed as x = (-b ± √(b² - 4ac)) / (2a). This formula provides the values of x that satisfy the equation, where b² - 4ac is known as the discriminant, which determines the nature of the roots.

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It indicates 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. Understanding the discriminant helps in predicting the solutions without solving the equation.