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

검증된 단계별 안내

Start with the given quadratic equation: $3x^2 - 2x - 2 = 0$.

Divide the entire equation by 3 to make the coefficient of $x^2$ equal to 1: $x^2 - \frac{2}{3}x - \frac{2}{3} = 0$.

Move the constant term to the right side: $x^2 - \frac{2}{3}x = \frac{2}{3}$.

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: $x^2 - \frac{2}{3}x + \frac{1}{9} = \frac{2}{3} + \frac{1}{9}$.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Solve each equation in Exercises 47–64 by completing the square. 3x2−2x−2=03x^2 - 2x - 2 = 0