Solve each equation in Exercises 47–64 by completing the square. $3x^2-2x-2=0$

Verified step by step guidance

1. Start with the given quadratic equation: \$3x^2 - 2x - 2 = 0$.
2. Divide the entire equation by 3 to make the coefficient of \(x^2\) equal to 1: \(x^2 - \frac{2}{3}x - \frac{2}{3} = 0\).
3. Move the constant term to the right side: \(x^2 - \frac{2}{3}x = \frac{2}{3}\).
4. To complete the square, take half of the coefficient of \(x\), which is \(-\frac{2}{3}\), divide by 2 to get \(-\frac{1}{3}\), then square it to get \(\left(-\frac{1}{3}\right)^2 = \frac{1}{9}\). Add this value to both sides: \(x^2 - \frac{2}{3}x + \frac{1}{9} = \frac{2}{3} + \frac{1}{9}\).
5. Rewrite the left side as a perfect square trinomial: \(\left(x - \frac{1}{3}\right)^2 = \frac{2}{3} + \frac{1}{9}\). Then simplify the right side by finding a common denominator.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves creating a binomial squared expression on one side, making it easier to solve for the variable by taking square roots.

Quadratic Equation Standard Form

A quadratic equation is typically written in the form ax² + bx + c = 0, where a, b, and c are constants. Understanding this form is essential for applying methods like completing the square, as it helps identify coefficients and organize terms properly.

Isolating the Variable

Isolating the variable involves rearranging the equation so that the variable term is alone on one side. In completing the square, this step is crucial to set up the equation for creating a perfect square trinomial and eventually solving for the variable.