Solve each equation in Exercises 83–108 by the method of your choice. 5x^2 + 2 = 11x

Key Concepts

Quadratic Equations

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. In the given equation, 5x^2 - 11x + 2 = 0 is derived by rearranging the original equation. Understanding how to manipulate and solve quadratic equations is essential for finding the values of x that satisfy the equation.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. For quadratic equations, this often involves finding two binomials that multiply to give the quadratic. Mastery of factoring techniques can simplify the solving process and provide solutions more efficiently.

The Quadratic Formula

The quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), provides a method for finding the roots of any quadratic equation. This formula is particularly useful when the equation cannot be easily factored. Understanding how to apply the quadratic formula allows students to solve for x in a systematic way, regardless of the complexity of the coefficients.