Solve each equation in Exercises 47–64 by completing the square. $x^{2} - 6 x - 11 = 0$

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Start with the given quadratic equation: $x^2 - 6x - 11 = 0$.

Move the constant term to the other side to isolate the $x$ terms: $x^2 - 6x = 11$.

To complete the square, take half of the coefficient of $x$, which is $-6$, divide by 2 to get $-3$, then square it to get $(-3)^2 = 9$.

Add this square (9) to both sides of the equation to maintain equality: $x^2 - 6x + 9 = 11 + 9$.

Rewrite the left side as a perfect square trinomial: $(x - 3)^2 = 20$.

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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 both sides to create a binomial squared, making it easier to solve for the variable.

Quadratic Equation Standard Form

A quadratic equation is typically written in the form ax² + bx + c = 0. Understanding this form helps identify coefficients needed for completing the square and applying the quadratic formula or other solving methods.

Solving Quadratic Equations

Solving quadratic equations means finding the values of the variable that satisfy the equation. Methods include factoring, completing the square, and using the quadratic formula, each useful depending on the equation's structure.

Solve each equation in Exercises 47–64 by completing the square. x2−6x−11=0x^2 - 6x - 11 = 0