Solve each equation using completing the square. x2 - 2x - 2 = 0

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 41

2장, 문제 41

Solve each equation using completing the square. x² - 2x - 2 = 0

검증된 단계별 안내

Start with the given quadratic equation: \(x^2 - 2x - 2 = 0\).

Move the constant term to the other side to isolate the quadratic and linear terms: \(x^2 - 2x = 2\).

To complete the square, take half of the coefficient of \(x\), which is \(-2\), divide by 2 to get \(-1\), then square it to get \((-1)^2 = 1\).

Add this square to both sides of the equation to maintain equality: \(x^2 - 2x + 1 = 2 + 1\).

Rewrite the left side as a perfect square trinomial: \((x - 1)^2 = 3\). From here, you can proceed by taking the square root of both sides to solve for \(x\).

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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 adding and subtracting a specific value to create a binomial squared, making it easier to solve for the variable.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. Understanding its structure is essential for applying methods like completing the square, factoring, or using the quadratic formula to find the roots.

Isolating the Variable

Isolating the variable involves rearranging the equation so that the variable term stands alone on one side. This step is crucial before completing the square, as it simplifies the process of forming a perfect square trinomial and solving the equation.